AN 


ELEMENTARY      TREATISE 


ON 


ASTRONOMY. 


IN    FOUR    PARTS 


CONTAINING 

A  SYSTEMATIC  AND  COMPREHENSIVE  EXPOSITION  OF  THE 

THEORY,  AND  THE  MORE  IMPORTANT  PRACTICAL 

PROBLEMS  ;  WITH  SOLAR,  LUNAR,  AND 

OTHER  ASTRONOMICAL  TABLES. 


IBtsisnth  {or  Use  ks  u  Etxt^lioo^  in  ^olleses  unii  ^tulJtmU«* 


BY   WILLIAM   A.  NORTON, 

LATE  PROFESSOR  OF  NATURAL  PHILOSOPHY  AND  ASTRONOMY 
IN  THE  UNIVERSITY  OF  THE  CITY  OF  NEW  YORK. 


PUBLISHED  BY 

WILEY  &  PUTNAM,  New  York  ;  THOMAS,  COWPERTHWAITE 
&  CO.,  Philadelphia  ;  C.  C.  LITTLE  &  CO.,  Boston, 

1839. 


Entered  according  to  an  Act  of  Congress,  in  the  year  1839,  by 

WILLIAM  A.  NORTON, 

in  the  Clerk's  Office  of  the  District  Court  of  the  Southern  District  of  New  York. 


J.  P.  Wright,  Printer.  74  Cedar  Street,  N.  Y. 


^^td^- 


Engineerinf!  & 
Sciences 


PREFACE. 


The  object  in  writing  the  present  treatise,  has  been 
to  provide  a  suitable  text  book  for  the  use  of  the  stu- 
dents of  Colleges  and  the  higher  Academies,  and  at  the 
same  time  to  furnish  the  practical  astronomer  with  rules 
or  formulae,  and  accurate  tables  for  performing  the  more 
important  astronomical  calculations. 

The  work  is  divided  into  four  Parts.    The  first  three 
Parts  contain  the  theory :  the  First  Part  treating  of  the 
determination  of  the  places  and  motions  of  the  hea- 
venly bodies ;  the  Second,  of  the  phenomena  result- 
ing from  the  motions  of  these  bodies,  and  of  their  ap- 
pearances, dimensions,  and  physical  constitution;  and 
the  Third,   of  the  theory  of  Universal   Gravitation. 
The  Fourth  Part  consists  of  practical  problems,  which 
^  are  solved  with  the  aid  of  the  tables  appended  to  the 
^  work.     An  Appendix  is  added,  containing  a  large  col- 
O  lection   of  useful  trigonometrical  formulae,   and  such 
\  investigations  of  astronomical  formulae  as,  from  theu' 
vf*  length,  could  not,   consistently  with  the  plan  of  the 
r  J^work,  be  admitted  into  the  text,  and  which  it  was  still 
deemed  advisable  to  retain  for  the  benefit  of  the  few 
who  might  wish  to  pursue  them. 


IV  PREFACE. 


The  chief  peculiarities  of  the  present  treatise  are, 
1.  The  adoption  of  the  Copernican  System  as  an  hy- 
pothesis at  the  outset,  leaving  it  to  be  established  by 
the  agreement  between  the  conclusions  to  which  it 
leads  and  the  results  of  observation.     2.  A  connected 
exposition  of  the  principles  and  methods  of  astronomi- 
cal observation,  embracmg  the  doctrine  of  the  sphere, 
the  construction  and  use  of  the  prmcipal  astronomical 
instruments,  and  the  theory  of  the  corrections  for  re- 
fraction, parallax,  aberration,  precession,  and  nutation. 
3.  The  exhibition  of  the  methods  of  determining  the 
motions  and  places  of  the  different  classes  of  the  hea- 
venly bodies  m  one  connection.     4.  The  explanation  of 
the  principles  of  the  construction  of  astronomical  tables. 
5.  The  addition  of  a  chapter  on  the  measurement  of 
time,  embracing  the  explanation  of  the  different  khids 
of  time,  the  processes  by  which  one  is  converted  into 
another,  the  methods  of  determhiing  the  time  from  as- 
tronomical observations  with  .the  transit  mstrument  and 
sextant,  and  the  calendar.     6.  The  contemplation  of 
the  phenomena  of  the  aspect  and  apparent  motions  of 
the  heavenly  bodies  as  consequences  of  theii*  motions  in 
space,  and  the  deduction  of  the  various  circumstances  i( 
of  these  phenomena  from  the  theory  of  the  orbitual  < 
motions  previously  established.     7.  A  comprehensive  '^ 
view  of  the  theory  of  Universal  Gravitation,  followed  /' 
out  into  its  various  consequences.    8.  An  exposition  of  '^ 
the  operations  of  the  disturbing  forces  in  producin^i 
the  perturbations  of  the  motions  of  the  Solar  System. 
9.  The  solution  of  practical  problems  by  means  of  loga- 


PREFACE. 


rithmic  formulse.  instead  of  rules.  10.  The  addition  of 
lunar,  solar,  and  other  astronomical  tables  of  pecuhar 
accuracy  and  improved  arrangement. 

It  may  further  be  mentioned,  that  many  of  the  inves- 
tigations have  been  materially  simplified,  and  that  the 
aim  ha§  been  to  introduce  into  all  of  them  as  much  sim- 
phcity  and  uniformity  of  method  as  possible.  Particu- 
lar attention  has  also  been  paid  to  the  diagrams,  it  being 
of  great  importance  that  they  should  convey  correct 
notions  to  the  mind  of  the  student. 

The  problems  in  the  Fourth  Part  are  principally  for 
making  calculations  relative  to  the  Sun,  Moon,  and  Fixed 
Stars.  The  tables  of  the  Sun  and  Moon,  used  in  finding 
the  places  of  these  bodies,  have,  for  the  most  part,  been 
abridged  and  computed  from  the  tables  of  Delambre, 
as  corrected  by  B'Cssel,  and  those  of  Burckhardt ;  and 
the  tables  of  epochs  have  all  been  reduced  to  the  meri- 
dian of  Greenwich.  These  tables  will  give  the  places 
and  motions  of  the  Sun  and  Moon  within  a  fraction  of  a 
second  of  the  tables  from  which  they  were  derived. 
But  as  this  degree  of  accuracy  will  not  generally  be 
required,  rules  are  also  given  m  the  Fourth  Part  for 
obtaining  approximate  results.  The  entire  set  of  tables 
has  been  stereotyped,  and  great  pains  has  been  taken, 
by  repeated  revisions  and  verifications,  to  render  them 
accurate. 

The  principal  astronomical  works  which  have  been 

'consulted  in  writing  the  present  treatise,  are  those  of 

Vince,  Gregory,  Woodhouse,  Delambre,  Biot,  Laplace, 

Herschel,  and  Gummere ;  also  Francceufs  Uranogra- 


PREFACE. 


pky,  FranccRufs  Practical  Astronomy,  Encyclopedia 
Metropolitana,  Article  Astronomy,  and  Bailey's  Tables 
and  Formidce.  Free  use  has  been  made  of  the  methods 
of  investigation  and  demonstration  pursued  in  these 
treatises,  such'  modifications  being  introduced,  in  those 
which  have  been  adopted,  as  the  plan  of  the  work 
required. 

A  list  of  the  Errata  that  have  been  detected,  which, 
it  is  beUeved,  contains  all  that  are  important,  may  be 
found  immediately  after  the  Table  of  Contents.  It  is 
recommended  to  the  student  to  make  some  note  of 
these  at  the  places  where  they  occiu-  m  the  text, 
before  taking  up  the  subject  in  course. 

WILLIAM  A.  NORTON. 


TABLE  OF  CONTENTS. 


PART    I. 

ON  THE  DETERMINATION  OF  THE   PLACES  AND 
MOTIONS  OF  THE  HEAVENLY  BODIES. 

CHAPTER  I. 

Page 
Introductory  Remarks — General  Phenomena  of  the  Heavens         ...       1 

CHAPTER  XL 

On  the  Celestial  and  Terrestrial  Spheres 8 

CHAPTER  III. 

On  the  Construction  and  Use  of  the  Principal  Astronomical  Instpuments      .  19 

Transit  Instrument       ..........  22 

Astronomical  Clock        ..........  27 

Astronomical  Quadrant         .........  28 

Altitude  and  Azimuth  Instrument        .......  31 

Equatorial     ............  ib. 

Sextant          ............  32 

Micrometer — Errors  of  Instrumental  Admeasurement  .        .        .        .34 

CHAPTER  IV. 

Theory  of  Corrections — Refraction — Parallax — Aberration — Precession — Nu- 
tation        ........        34 

Refraction 35 

Parallax         ............  41 

Aberration 47 

Precession  and  Nutation 52 

Remarks  on  the  Corrections. — Verification  of  the  Hypothesis  that  the 
Diurnal  Motion  of  the  Stars  is  Uniform  and  Circular       .         .        .59 


Vlll  TABLE    OF    CONTENTS. 

CHAPTER  V. 

Page 
Of  the  Earth  ; — its  Figure  and  Dimensions ; — Latitude  and  Longitude  of  a 

Place 61 

Determination  of  the  Latitude  and  Longitude  of  a  Place     .         .         .64 

CHAPTER  VI. 

Of  the  Places  of  the  Fixed  Stars 67 

CHAPTER  VH. 

Of  the  Apparent  Motion  of  the  Sun  in  the  Heavens 72 

CHAPTER  Vni. 

Of  the  Motions  of  the  Sun,  Moon,  and  Planets,  in  their  Orbits      .         ,         .77 

Keplefs  Laws ib. 

Definitions  of  Terms      ..........     81 

Elements  of  the  Orbit  of  a  Planet         .         .         .         .         .         .         .82 

Methods  of  Determining  the  Elements  of  the  Sun's  Apparent  Orbit,  or 
of  the  Earth's  Real  Orbit    .........     83 

Methods  of  Determining  the  Elements  of  the  Moon's  Orbit  .         .     88 

Methods  of  Determining  the  Elements  of  a  Planet's  Orbit  .         .     91 

Mean  Elements  and  their  Variations     .......     98 

CHAPTER  IX. 

On  the  Determination  of  the  Place  of  a  Planet,  or  of  the  Sun,  or  Moon,  for 
a  Given  Time,  by  the  Elliptical  Theoiy ;  and  of  the  Verification  of  Kep- 

ler's  Laws 100 

Place  of  a  Planet,  or  of  the  Sun  or  Moon  in  its  Orbit          .         .         .  ib. 

Heliocentric  Place  of  a  Planet     ........  102 

Geocentric  Place  of  a  Planet       ........  ib. 

Places  of  the  Sun  and  Moon         .         .         .         .         .         .         .         .104 

Verification  of  Kepler's  Laws 105 

CHAPTER  X. 

On  the  Inequalities  of  the  Motions  of  the  Planets  and  of  the  Moon ;  and  of 
the  Construction  of  Tables  for  finding  the  Places  of  these  Bodies     .         .     106 
Construction  of  Tables 112 

CHAPTER  XI. 

Of  the  Motions  of  the  Comets 117 

CHAPTER  XII. 

Of  the  Motions  of  the  Satellites 121 


TABLE    OF    CONTENTS.  iX 

CHAPTER  XIII. 

Page 

On  the  Measurement  of  Time 124 

Different  Kinds  of  Time ib. 

Conversion  of  one  Species  of  Time  into  another  ....     126 

Determination  of  the  Time  and  Regulation  of  Clocks  by  Astronomical 
Observations     .         .         .         .         .         .         •         •         •         •         .128 

Of  the  Calendar 132 


PART    II. 

ON  THE  PHENOMENA  RESULTING  FROM  THE  MOTIONS 
OF  THE  HEAVENLY  BODIES,  AND  ON  THEIR  APPEAR. 
ANCES,  DIMENSIONS,  AND  PHYSICAL  CONSTITUTION. 

CHAPTER  XIV. 

Page 
Of  the  Sun  and  the  Phenomena  attending  its  Apparent  Motions  .         .     137 

Inequality  of  Days ib. 

Twilight 141 

The  Seasons 142 

Appearance,  Dimensions,  and  Physical  Constitution  of  the  Sun         .     145 

CHAPTER  XV. 

Of  the  Moon  and  its  Phenomena 149 

Phases  of  the  Moon ib. 

Moon's  Rising,  Setting,  and  Passage  over  the  Meridian      .        ,        .  152 

Rotation  and  Librations  of  the  Moon            ......  155 

Dimensions  and  Physical  Constitution  of  the  Moon     ....  156 

CHAPTER  XVI. 

Eclipses  of  the  Sun  and  Moon — Occultations  of  the  Fixed  Stars         .        .158 

Eclipses  of  the  Moon ib. 

Eclipses  of  the  Sun 168 

Occultations 186 

CHAPTER  XVII. 

Of  the  Planets  and  the  Phenomena  occasioned  by  their  Motions  in  Space  .    187 
Apparent  Motions  of  the  Planets  with  respect  to  the  Sun    .        .        .      ib* 

0 


X  TABLE    OF    CONTENTS. 

Page 

Stations  and  Re! I  ngradations  of  the  Planets 191 

Phases  of  the  Inferior  Planets     ........  193 

Transits  of  the  Inferior  Planets           .......  194 

Appearances,  Dimensions,  Rotation,  and  Physical  Constitution  of  the 

Planets ' 195 

CHAPTER  XVIII. 

Of  Comets. — Their  Appearance,  Magnitude,  and  Physical  Constitution       .  202 

CHAPTER  XIX. 

Of  the  Fixed  Stars. — Their  Number  and   Distribution   over  the  Heavens — 
Annual   Paralhix   and  Distance — Variable  Stars — Double  Stars — Clusters 

of  Stars,  and  Nebulas 204 

Annual  Parallax  and  Distance  of  the  Stars 205 

Variable  Stars 206 

Double  Stars 208 

Clusters  of  Stars— Nebula 209 

i 


PART   III. 

OF  THE  THEORY  OF  UNIVERSAL  GRAVITATION, 
CHAPTER  XX. 

Page 
Of  the  Principle  of  Universal  Gravitation 211 

CHAPTER  XXI. 

Theory  of  the  Elliptic  Motion  of  the  Planets 213 

CHAPTER  XXII. 

Theory  of  the  Perturbations  of  the  Elliptic  Motion  of  the  Planets  and  of  the 
Moon 218 

CHAPTER  XXIII. 

Of  the  Relative  Masses  and  Densities  of  the  Sun,  Moon,  and  Planets;  and 
of  the  Relative  Intensity  of  the  Gravity  at  tlieir  surface  >o         .         ,         .    235 


TABLE    OF    CONTENTS.  XI 

CHAPTER  XXIV. 

Page 
Of  the  Figure  and  Rotation  of  the  Earth  ;  and  of  the  Precession  of  the  Equi- 
noxes  and  Nutation 236 

CHAPTER  XXV. 

Of  the  Tides 240 


PART   IV. 

ASTRONOMICAL  PROBLEMS. 

Page 

Explanation  of  the  Tables 247 

Prob.  I.     To  work,  by  logistical  logarithms,  a  proportion  the  terms  of  which 

are  degrees  and  minutes,  or  minutes  and  seconds  of  an  arc  ;  or  hours 

and  minutes,  or  minutes  and  seconds  of  time 254 

Prob.  II.     To  take  from  a  table  the  quantity  corresponding  to  a  given  value 

of  the  argument,  or  to  given  values  of  the  arguments  of  the  table  .  255 
Prob.  III.     To  convert  Degrees,  Minutes,  and  Seconds  of  the  Equator  into 

Time 262 

Prob.  IV.     To  convert  Time  into  Degrees,  Minutes,  and  Seconds       .         .     262 

Prob.  V.     The  Longitudes  of  two  Places,  and  the  Time  at  one  of  them  being 

given,  to  find  the  corresponding  time  at  the  other      ....     262 

Prob.  VI.     The  Apparent  Time  being  given,  to  find  the  corresponding  Mean 

Time  ;  or  the  Mean  Time  being  given,  to  find  the  Apparent      .         .     264 

Prob.  VII.  To  correct  the  Obsei-ved  Altitude  of  a  Heavenly  Body  for  Re- 
fraction      267 

Prob.  VIII.    The  Apparent  Altitude  of  a  Heavenly  Body  being  given,  to  find 

its  True  Altitude .        .        .269 

Prob.  IX.     To  find  the  Sun's  Longitude,  Semi-diameter,  and  Hourly  Motion, 

for  a  given  Time,  from  the  Tables 271 

Prob.  X.     To  find  the  Apparent  Obliquity  of  the  Ecliptic,  for  a  given  Time, 

from  the  Tables 274 

Prob.  XI.     Given  the  Sun's  Longitude  and  the  Obliquity  of  the  Ecliptic,  to 

find  his  Right  Ascension  and  Declination  .....     275 

Prob.  XII.     Given  the  Sun's  Right  Ascension  and  the  Obliquity  of  the  Eclip- 

tic,  to  find  his  Longitude  and  Declination  ......    276 


Xll  TABLE    OF    CONTENTS. 

Page 
Prob.  XIII.     The  Sun's  Longitude  and  the  Obliquity  of  the  Ecliptic  being 

given,  to  find  the  Angle  of  Position 277 

Pkob.  XIV,  To  find  from  the  Tables,  the  Moon's  Longitude,  Latitude,  Equa- 
torial  Parallax,  Semi-diameter,  and  Hourly  Motions  in  Longitude  and 
Latitude,  for  a  given  Time •         •     277 

Prob.  XV,     The  Moon's  Equatorial  Parallax,  and  the  Latitude  of  a  Place 

being  given,  to  find  the  Reduced  Parallax  and  Latitude     .         .         .     288 

Prob,  XVI.     To  find  the  Longitude  and  Altitude  of  the  Nonagesimal  Degree 

of  the  Ecliptic,  for  a  given  Time  and  Place 289 

Prob.  XVII.  To  find  the  Apparent  Longitude  and  Latitude,  as  afl^ected  by 
Parallax,  and  the  Augmented  Semi-diameter  of  the  Moon  ;  the  Moon's 
True  Longitude,  Latitude,  Horizontal  Semi  diameter,  and  Equatorial 
Parallax,  and  the  Longitude  and  Altitude  of  the  Nonagesimal  Degree 
of  the  Ecliptic,  being  given        ....,,..     292 

Prob.  XVIII.  To  find  the  Mean  Right  Ascension  and  Declination,  or  Longi- 
tude and  Latitude  of  a  Star,  for  a  given  Time,  from  the  Tables  .     297 

Prob,  XIX.  To  find  the  Aberration  of  a  Star  in  Right  Ascension  and  Decli- 
nation, for  a  given  Day      299 

Prob,  XX.  To  find  the  Nutation  of  a  Star  in  Right  Ascension  and  Declina- 
tion, for  a  given  Day  ,..,,...,     300 

Prob.  XXI.     To  find  the  Apparent  Right  Ascension  and  Declination  of  a  Star 

for  a  given  Day  .,....,..,     301 

Prob.  XXII.     To  find  the  Aberration  of  a  Star  in  Longitude  and  Latitude,  for 

a  given  Day      ...........     302 

Prob.  XXIII.     To  find  the  Apparent  Longitude  and  Latitude  of  a  Star,  for  a 

given  Day 303 

Prob.  XXIV.  To  compute  the  Longitude  and  Latitude  of  a  Heavenly  Body 
from  its  Right  Ascension  and  Declination,  the  Obliquity  of  the  Eclip. 
tic  being  given 304 

Prob.  XXV.  To  compute  the  Right  Ascension  and  Declination  of  a  Heavenly 
Body  from  its  Longitude  and  Latitude,  the  Obliquity  of  the  Ecliptic 
being  given 395 

Prob.  XXVI.     The  Longitude  and  Declination  of  a  Body  being  given,  and 

also  the  Obliquity  of  the  Ecliptic,  to  find  the  Angle  of  Position  .     307 

Prob.  XXVII.     To  find  from  the  Tables  the  Time  of  New  or  Full  Moon,  for 

a  given  Year  and  Month    .........     308 

Prob.  XXVIII.  To  determine  the  number  of  Eclipses  of  the  Sun  and  Moon 
that  may  be  expected  to  occur  in  any  given  Year,  and  the  Times 

nearly  at  which  they  will' take  place 01 2 

Prob.  XXIX.     To  calculate  an  Eclipse  of  the  Moon 315 

Prob.  XXX.    To  calculate  an  Eclipse  of  the  Sun,  for  a  given  Place   .  330 


TABLE    OF   CONTENTS.  Xiii 

APPENDIX. 

Page 

Trigonometrical  Formulae .  341 

I.  Relative  to  a  Single  arc  or  angle  a ib. 

II.  Relative  to   Two  Arcs  a  and  b,  of  which  a  is  supposed  to  be  the 
greater           . 342 

III.  Trigonometrical  Series 343 

IV.  Differences  of  Trigonometrical  Lines ib. 

V.  Resolution  of  Right  Angled  Spherical  Triangles        .         .         .  344 

VI.  Resolution  of  Oblique  Angled  Spherical  Triangles   .        .         .  346 
Investigation  of  Astronomical  FoRMULiE 349 

Formula  for  the  Parallax  in  Right  Ascension  and  Declination,  and 

in  Longitude  and  Latitude             ib. 

Formula  for  the  Aberration  in  Longitude  and  Latitude,  and  in  Right 

Ascension  and  Declination     .......  357 

Formula  for  the  Nutation  in  Right  Ascension  and  Declination         .  362 

Solution  of  Kepler's  Problem,  by  which  a  Body's  Place  is  found  in  an 

Flliptical  Orbit             367 

Note  to  Problem  XIV        ..........  37I 

Rules  for  finding  the  Moon's  Longitude,  Latitude,  Hourly  Motions, 
Equatorial  Parallax,  and  Semi.diameter,for  a  given  time,  from 

the  Nautical  Almanac jb. 


0*0: 


ERRATA. 


Page  31,  line  12.  For  S  d,  read  T  a. 

"     44.  i^or  tang  \{z  ^  p),  read  tang  {^  z  -{-  p).  • 

"     ib.  For  z,  read  ^  z. 

"    ib.  For  0  s',  read  o  s. 

"     58,  "     14.  For  s  a',  read  sin  s  a'. 

«     63,  "      4.  For  Art.  132,  read  Art.  142. 

«     69,  "    22.  For  Table  LXl,  rea</  Table  XC. 

"     73,  "     10.  jPor  declination,  read  right  ascension. 

«    84,  «     31.  For  Art.  179,  read  Art.  178. 

«    ib.,  "     36.  For  Art.  174,  read  Art.  184. 

"  87,  "  28.  For  motion  of  the  apogee  and  perigee  in  lon- 
gitude, read  sidereal  motion  of  apogee  and 
perigee. 

103.  For^—:-\read]-'^''^{ 

1  +  r  1  +  tang  6 

105,  "  13.  For  246,  read  247. 

116,  "  6.  For  Art.  265,  read  Art.  267. 

119,  ''  20.  For  1836,  read  1826. 

131,  "  4.  For  4  20,    read    3  50.     This  ivill   change 

sligJttly  the  result  of  the  Problem. 

145.  For  tang  i  A  E  B,  read  sin  ^  A  E  B. 

ib.  For  tang  \  S,  read  sin  ^  S. 

155,  "  3.  For  m  —  S,  read  m  —  s. 

iKr      ::  1    .     Et     61' 24"  ,    33' 31" 

156,  "last  For  ^^^,rea.^j^^^, 

159.  For  tang  E  C  a,  reac?  sin  E  C  a. 
ib.  For  tang  (5  —  p),  reacZ  sin  {5  —  p). 
ib.,     "     26.  For  tangent,  read  sine. 

160,  "     25.  For  maximum,  read  minimum,  and/or  mini- 
mum, read  maximum. 


XVI  ERRATA. 

Page  165,  line  13.  For  C'f,  read  M/,  a7idfor  C'/',  read  M/'. 
"  167.  For  C  R,  read  C  R'. 

"  178,     "     16.  For  Prob.  XXVII,  read  Prob.  XVII. 
"  183,  equ.121.  For  3600%  read —3600': 
«     ib.,  lastequ.  For  1800^-,  read  —  1800^-. 
"  185,  line  17.  For  M  m,  read  M  w. 
"  200,     "    25.  For  1848,  read  1847. 

"  205,     "     33.  For  diametrically  to,  read  diametrically  op- 
posite to. 
"  215,     "       6.  For  conclave,  read  concave. 
"  220,  last  equ.  For  cos  7,  read  cos  /3. 

"  225,  line  20.  For  apogee  to  the  perigee,  read  perigee  to  the 
apogee, 
19.  For  (equa.  137),  read  (equa.  139). 
14.  For  Art.  563,  read  Art.  564. 
"13(fcl9.  For  S,  read  1^. 

16.  For  the  same  of,  read  the  same  side  of. 

6.  For  ^lo,  read  ^ij. 
21.  For  LVII,  read  LXVII. 
23.  For  logistical  of,  read  logistical  logarithm  of. 

7.  For  XVII,  read  XVIII. 
For  last  Problem,  read  Problem  VI. 

5.  For  XXVII,  reacZ  LIV. 

6.  For  —  2'  37".2,  read  237". 2. 
1.  For  R',  read  R. 

17.  i^or  approximate  of,  read  approximate  time  of. 
For  X,  read  X'. 

"  348,   case  2.  i^or  A,  c,  6,  reac?  A,  C,  b,  and  for  C,  reac?  c. 

"  350,  2d  equ.  For  s  p  s',  read  z  p  s'. 

«  354.  For  tang  (5  —  90°),  read  tang  |(s  —  90°). 


«  228, 

ii 

«  229, 

u 

"  230, 

u 

"    ib.. 

u 

«  238, 

a 

«  252, 

a 

«  254, 

a 

"  265, 

u 

"  271. 

«  279, 

u 

«  281, 

a 

"  293, 

c: 

"  309, 

li 

«  319. 

PART    I. 


ON  THE  DETERMINATION  OF  THE  PLACES  AND 
MOTIONS  OF  THE  HEAVENLY  BODIES. 


CHAPTER    I. 

INTRODUCTORY    REMARKS. GENERAL    PHENOMENA    OP 

THE     HEAVENS. 

1.  Astronomy  is  a  mixed  mathematical  science,  which  treats 
of  the  motions,  positions,  distances,  appearances,  magnitudes, 
and  physical  constitution  of  the  heaA'enly  bodies.  That  part  of 
the  science,  which  has  for  its  object  the  determination  of  these 
several  particulars  from  observation,  is  called  Plane  Astronomy ; 
and  that,  in  which  the  physical  causes  of  the  motions  and  con- 
stitution of  the  heavenly  bodies  are  investigated,  is  denominated 
Physical  Astronomy. 

2.  To  be  able  to  form  correct  notions  of  the  phenomena  of 
the  heavens,  it  is  necessary  to  know  the  form  of  the  earth.  We 
learn  from  the  following-  circumstances,  that  the  earth  is  a  body 
of  a  globular  form,  insulated  in  space.  1st.  When  a  vessel  is 
receding  from  the  land,  an  observer,  stationed  upon  the  coast, 
first  loses  sight  of  the  hull,  then  of  the  loAver  parts  of  the  sails^ 
and  lastly,  of  the  top-sails.  This  is  the  case,  whatever  is  the 
direction  of  the  course  of  the  vessel,  and  at  whatever  part  of 
the  earth  it  is  observed.  2d.  At  sea,  the  visible  horizon,  or  the 
line  bounding  the  visible  portion  of  the  earth's  surface,  is  every 
where  a  circle,  of  a  greater  or  less  extent,  according  to  the 
altitude  of  the  point  of  observation,  and  is,  on  all  sides,  equally 
depressed.     3d.  Navigators  have  sailed  around  the  earth,  and, 

1 


3  ASTRONOMY. 

by  steering  their  course  continually  in  one  direction,  arrived,  at 
length,  at  the  place  from  which  they  departed.  These  facts 
prove  the  surface  of  the  sea  to  be  convex,  and  the  surface  of 
the  land  conforms  very  nearly  to  that  of  the  sea ;  for,  the  eleva- 
tions of  the  highest  mountains  above  the  level  of  the  ocean  bear 
an  exceedingly  small  proportion  to  the  dimensions  of  the  whole 
earth. 

3.  If,  on  a  clear  night,  we  observe  the  heavens,  we  shall  find 
that  the  stars,  while  they  retain  the  same  situations  with  respect  to 
each  other,  undergo  a  continual  change  of  position  with  respect 
to  the  earth.  Some  will  be  seen  to  ascend  from  a  quarter  called 
the  East^  being  replaced  by  others  that  come  into  view,  or  rise  ; 
others,  to  descend  towards  the  opposite  quarter,  the  West,  and  to 
go  out  of  view,  or  set.  And,  if  our  observations  be  continued 
throughout  the  night,  with  the  east  on  our  left  and  the  west  on 
our  right,  the  stars  which  rise  in  the  east  will  be  seen  to  move 
in  parallel  circles  entirely  across  the  visible  heavens,  and  finally 
to  set  in  the  west.  Each  star  will  ascend  in  the  heavens  during 
the  first  half  of  its  course,  and  descend  during  the  remaining 
half  The  greatest  heights  of  the  several  stars  will  be  differ- 
ent, but  they  will  all  be  attained  towards  that  part  of  the  heavens 
which  lies  directly  in  front,  called  the  ^onth.  If  we  now  turn 
our  backs  to  the  south,  and  direct  our  attention  to  the  opposite 
quarter,  the  North,  new  phenomena  will  present  themselves. 
Some  stars  will  appear,  as  before,  ascending,  reaching  their 
greatest  heights,  and  descending ;  but  other  stars  v\dll  be  seen, 
farther  to  the  north,  that  never  set,  and  which  appear  to  revolve 
in  circles,  from  east  to  west,  about  a  certain  star,  that  seems  to 
remain  stationary.  This  seemingly  stationary  star  is  called  the 
Polar  Star  ;  and  those  stars  that  revolve  about  it,  and  never  set, 
are  called  Circum-polar  Stars.  It  should  be  remarked,  how- 
ever, that  the  polar  star,  when  accurately  observed  by  means  of 
instruments,  is  found  not  to  be  strictly  stationary,  but  to  describe 
a  small  circle  about  a  point  at  a  little  distance  from  it,  as  a  fixed 
centre.  This  point  is  called  the  North  Pole.  It  is,  in  reality, 
about  the  north  pole,  as  thus  defined,  and  not  the  polar  star, 
that  the  apparent  revolutions  of  the  stars  at  the  north  are  per- 
formed. At  the  corresponding  hours  of  the  following  night, 
the  aspect  of  the  heavens  will  be  the  same,  from  which  it 


GENERAL  PHENOMENA  OP  THE  HEAVENS.         3 

appears  that  the  stars  return  to  the  same  position  once  in  about 
24  hours.  It  would  seem,  then,  that  the  stars  all  appear  to 
move,  from  east  to  west,  exactly  as  if  attached  to  the  concave 
surface  of  a  hollow  sphere,  which  rotates  in  this  direction  about 
an  axis  passing  through  the  station  of  the  observer  and  the  north 
pole  of  the  heavens,  in  a  space  of  time  nearly  equal  to  24  hours. 
This  motion,  common  to  all  the  heavenly  bodies,  is  called  their 
Diurnal  Motion. 

4.  It  is  ascertained  by  certain  accurate  methods  of  observation 
and  computation,  hereafter  to  be  exhibited,  that  the  diurnal  mo- 
tion of  the  stars  is  strictly  uniform  and  circular . 

5.  A  circle  cut  out  of  the  heavens  by  a  plane  passing  through 
the  axis  of  rotation,  has  a  north  and  south  direction  ;  and  a  cir- 
cle cut  out  by  a  plane  perpendicular  to  the  axis,  has  an  east  and 
west  direction. 

6.  The  greater  number  of  the  stars  preserve  constantly  the 
same  relative  position  with  respect  to  each  other ;  and  they  are 
therefore  called  Fixed  Stars.  There  are,  however,  a  few  stars, 
called  Planets.,  which  are  perpetually  changing  their  places  in 
the  heavens.  The  number  of  the  planets  is  ten.  Each  has  a 
distinctive  name,  as  follows :  Mercury,  Venus,  Mars,  Jupiter, 
Saturn,  Uranus,  Ceres,  Pallas,  Juno,  and  Vesta.  Mercury, 
Venus,  Mars,  Jupiter  and  Saturn,  are  visible  to  the  naked  eye, 
and  have  been  known  from  the  most  ancient  times.  The  other 
five,  namely,  Uranus,  Ceres,  Pallas,  Juno,  and  Vesta,  cannot  be 
seen  without  the  assistance  of  the  telescope,  and  were  discovered 
by  modern  observers.* 

7.  The  planets  are  distinguishable  from  each  other,  either  by 
a  difference  of  aspect,  or  by  a  difference  of  apparent  motion  with 
respect  to  the  sun.  Venus  and  Jupiter  are  the  two  most  brilliant 
planets :  they  are  quite  similar  in  appearance,  but  their  appa- 
rent motions,  with  respect  to  the  sun,  are  very  different.  Venus 
never  recedes  beyond  40°  or  50°  from  the  sun,  while  Jupiter  is 


*  The  planet.  Uranus  was  discovered  in  1781  by  Dr.  Herschel,  who  gave  il  the 
name  of  tlie  Georgium  Sidua.  By  the  European  astronomers  it  was  called 
Herschel.  It  is  now  generally  known  by  the  name  given  in  the  text.  Ceres, 
Pallas,  Juno,  and  Vesta,  have  been  disC'verod  since  1800  ;  the  first  by  Piazzi,  the 
second  and  fourth  by  Olbers,  and  the  third  by  Harding. 


4  ASTRONOMY. 

seen  at  every  variety  of  angular  distance  from  him.  Mars  is 
known  by  the  ruddy  color  of  his  light.  Saturn  has  a  pale,  dull 
aspect. 

8.  The  apparent  motion  of  the  planets  is  generally  directed 
towards  the  east ;  occasionally,  however,  they  are  seen  moving 
towards  the  west.  As  their  easterly  prevails  over  their  westerly 
motion,  they  all,  in  process  of  time,  accomplish  a  revolution 
around  the  earth.  The  periods  of  revolution  are  different  for 
each  planet. 

9.  The  Sun  and  Moon  are  also  continually  changing  their 
places  among  the  fixed  stars. 

10.  From  repeated  examinations  of  the  situation  of  the  moon 
among  the  stars,  it  is  found,  that  she  has  with  respect  to  them  a 
progressive  circular  motion,  from  ivest  to  east,  and  completes  a 
revolution  around  the  earth  in  about  27  days. 

11.  The  motion  of  the  sun  is  also  constantly  progressive,  and 
directed  from  west  to  east.  This  will  appear,  on  observing  for 
a  number  of  successive  evenings,  the  stars  which  first  become 
visible  in  that  part  of  the  heavens  where  the  sun  sets.  It  will 
be  found,  that  those  stars,  which  in  the  first  instance  were  ob- 
served to  set  just  after  the  sun,  soon  cease  to  be  visible,  and  are 
replaced  by  others  that  were  seen  immediately  to  the  east  of 
them ;  and  that  these,  in  their  turn,  give  place  to  others  situated 
still  farther  to  the  east.  The  sun,  then,  is  continually  approach- 
ing the  stars  that  lie  on  the  eastern  side  of  him.  The  period  of 
time,  in  which  he  accomplishes  a  revolution  in  the  heavens,  is 
about  365  days. 

It  is  to  be  observed  that  the  sun  does  not  advance  directly 
towards  the  east.  He  has  also  some  motion  from  south  to 
north,  and  north  to  south.  It  is  a  matter  of  common  observa- 
tion, that  the  sun  is  moving  towards  the  north  from  winter  to 
summer,  and  towards  the  south  from  summer  to  winter. 

12.  When  the  place  of  the  sun  in  the  heavens  is  accurately 
found  from  day  to  day  by  certain  methods  of  observation, 
hereafter  to  be  explained,  it  appears  that  his  path  is  an 
exact  circle,  inclined  about  23°  to  a  circle  running  due  east 
and  west.     (Art.  .5.) 

13.  The  motions  of  the  sun,  moon,  and  planets,  are  for  the 
most  part  confined  to  a  certain  zone,  of  about  18°  in  breadth, 


GENERAL  PHENOMENA  OF  THE  HEAVENS.         5 

extending  around  the  heavens  from  west  to  east,  which  has 
received  the  name  of  the  Zodiac. 

14.  There  is  yet  another  class  of  bodies,  called  Comets,  that 
have  a  motion  among  the  fixed  stars.  They  appear  only  occa- 
sionally in  the  heavens,  and  continue  visible  only  for  a  few 
weeks,  or  months.  They  shine  with  a  diffusive,  nebulous  light, 
and  are  commonly  accompanied  by  a  faint  divergent  stream  of 
similar  light,  called  a  tail. 

15.  The  motions  of  the  comets  are  not  restricted  to  the  zodiac. 
These  bodies  are  seen  in  all  parts  of  the  heavens,  and  moving 
in  every  variety  of  direction. 

16.  By  inspecting  the  planets  with  telescopes,  it  has  been 
discovered  that  some  of  them  are  constantly  attended  by  a 
greater  or  less  number  of  small  stars,  whose  positions  are 
continually  varying.  These  attendant  stars  are  called  Satel- 
lites. The  planets  which  have  satellites  are  Jupiter,  Saturn, 
and  Uranus.  The  satellites  are  sometimes  called  Secondary 
Planets  ;  the  planets  upon  which  they  attend  being  denomi- 
nated Primary  Planets. 

17.  The  sun  and  moon,  the  planets,  (including  the  earth,) 
together  with  their  satellites,  and  the  comets,  compose  the  Solar 
System. 

18.  From  the  consideration  of  the  apparent  motions  and 
other  phenomena  of  the  Solar  System,  several  theories  have 
been  formed  in  relation  to  the  arrangement  and  actual  motions 
in  space  of  the  bodies  that  compose  it.  The  theory,  or  system, 
now  universally  received,  is  (in  its  most  prominent  features) 
that  which  was  taught  by  Copernicus  in  the  sixteenth  century, 
and  which  is  known  by  the  name  of  the  Copernican  System. 
It  is  as  follows  : 

19.  The  sun  occupies  a  fixed  centre,  about  which  the  planets 
(including  the  earth)  revolve  from  west  to  east,*  in  planes  that  are 
but  slightly  inclined  to  each  other,  and  in  the  following  order: 
Mercury,  Venus,  the  Earth,  Mars,  Vesta,  Juno,  Ceres,  Pallas, 
Jupiter,  Saturn,  and  Uranus.  The  earth  rotates  from  west  to 
east  about  an  axis,  inclined  to  the  plane  of  its  orbit,  under  an 


*  A  motion  in   space  from  West  to  East,  is  a  motion  from  right  to  left,  to  a 
person  situated  within  the  orbit  described,  and  on  tlie  north  side  of  its  plane. 


6  ASTRONOMY. 

angle  of  about  60^°,  and  wliich  remains  continually  parallel  to 
itself  as  the  earth  revolves  around  the  sun.  The  luoou  revolves 
from  west  to  east  around  the  earth  as  a  centre  ;  and,  in  like  man- 
nar,  the  satellites  circulate  from  west  to  east  around  their  prima- 
ries. Without  the  Solar  System,  and  at  immense  distances  from 
it,  are  the  fixed  stars.  (See  Plate  1,  which  is  a  diagram  of 
the  Solar  System  in  projection.) 

20.*  We  shall  here,  at  the  outset,  adopt  this  system  as  an 
hypothesis^  and  shall  rely  upon  the  simple  and  complete  expla- 
nations it  affords  of  the  celestial  phenomena,  as  they  come  to  he 
investigated,  together  with  the  evidence  furaished  by  Physical 
Astronomy,  to  produce  entire  conviction  of  its  truth  in  the  mind 
of  the  student. 

21.  The  following  are  the  characters  or  symbols  employed  by 
astronomers  for  denoting  the  several  planets,  and  the  sun  and 
moon : — 

The  Sun,     .     .     .     .   O         Ceres, ? 

Mercury, $         Pallas, 0 

Venus, 9         Jupiter, % 

The  Earth,  ....  ©         Saturn, \i 

Mars, $         Uranus, M 

Vesta, S         The  Moon,  ....  5 

Juno, 0 

22.  The  angular  distance  between  any  two  fixed  stars  is 
found  to  be  the  same,  from  whatever  point  on  the  earth's  surface 
it  is  measured.  It  follows,  therefore,  that  the  diameter  of  the 
earth  is  insensible,  when  compared  with  the  distance  of  the  fixed 
stars ;  and  that,  with  respect  to  the  rea^ion  of  space  which  sepa- 
rates us  from  these  bodies,  the  whole  earth  is  a  mere  point. 
Moreover,  the  angular  distance  between  any  two  fixed  stars  is 
the  same,  at  whatever  period  of  the  year  it  is  measured.  Whence, 
if  the  earth  revolves  around  the  sun,  its  entire  orbit  must  be 
insensible,  in  comparison  with  the  distance  of  the  stars. 

23.  On  the  hypothesis  of  the  earth's  rotation,  the  diurnal 
motion  of  the  heavens  is  a  mere  illusion,  occasioned  by  the 
rotation  of  the  earth.  To  explain  this,  suppose  the  axis  of  the 
earth  prolonged  on  till  it  intersects  the  heavens,  considered  as 
concentric  with  the  earth.  Conceive  a  g-reat  circle  to  be  traced 
through  the  two  points  of  intersection  and  the  point  directly 


GENERAL  PHENOMENA  OP  THE  HEAVENS.         7 

over  head,  and  let  the  position  of  the  stars  be  referred  to  this 
circle.  It  will  be  readily  perceived,  that  the  relative  motion  of 
this  circle  and  the  stars  will  be  the  same,  whether  the  circle 
rotates  with  the  earth  from  west  to  east,  or,  the  earth  being  sta- 
tionary, the  whole  heavens  rotate  about  the  same  axis  and  at 
the  same  rate  in  the  opposite  direction.  Now,  as  the  motion  of 
the  earth  is  perfectly  equable,  we  are  insensible  of  it,  and,  there- 
fore, attribute  the  changes  in  the  situations  of  the  stars,  with 
respect  to  the  earth,  to  an  actual  motion  of  these  bodies.  It 
follows,  then,  that  we  must  conceive  the  heavens  to  rotate  as 
above  mentioned,  since,  as  we  have  seen,  such  a  motion  would 
give  rise  to  the  same  changes  of  situation  as  the  supposed  rota- 
tion of  the  earth.  It  was  stated  (Art.  3)  that  the  sphere  of  the 
heavens  appears  to  rotate  about  a  line  passins;  through  the  north 
pole  and  the  station  of  the  observer ;  but,  as  the  radius  of  the 
earth  is  insensible  in  comparison  with  the  distance  of  the  stars, 
an  axis  passing  through  the  centre  of  the  earth  will,  in  appear- 
ance, pass  through  the  station  of  the  observer,  wherever  it  may 
be  upon  the  earth's  surface. 

24.  We  in  like  manner  infer,  that  the  observed  motion  of  the 
sun  in  the  heavens  is  only  an  apparent  motion,  occasioned  by  the 
orbitual  motion  of  the  earth.  Let  E  E'  (Fig.  1)  represent  two  po- 
sitions of  the  earth  in  its  orbit  E  E'  E"  about  the  sun  S.  When 
the  earth  is  at  E,  the  observer  will  refer  the  sun  to  that  part  of 
the  heavens  marked  5 ;  but  when  the  earth  is  arrived  at  E',  he 
will  refer  it  to  the  part  marked  s' ;  and  being  in  the  mean  time 
insensible  of  his  own  motion,  the  sun  will  appear  to  him  to  have 
described  in  the  heavens  the  arc  ss',  just  the  same  as  if  it  had 
actually  passed  over  the  arc  S  S'  in  space,  and  the  earth  had, 
during  that  time,  remained  quiescent  at  E.  The  motion  of  the 
sun  from  .<?  towards  s'  will  be  from  west  to  east,  since  the  motion 
of  the  earth  from  E  towards  E'  is  in  this  direction.  Moreover, 
as  the  axis  of  the  earth  is  inclined  to  the  plane  of  its  orbit  under 
an  angle  of  66i°,  (Art.  19,)  the  plane  of  the  sun's  apparent  path, 
which  is  the  same  as  that  of  the  earth's  orbit,  will  be  inclined 
23|°  to  a  circle  perpendicular  to  the  earth's  axis,  or  to  a  circle 
directed  due  east  and  west. 


ASTRONOMY. 


CHAPTER     II. 

ON    THE    CELESTIAL    AND    TERRESTRIAL    SPHERES. 

25.  In  determining  from  observation  the  apparent  positions 
and  motions  of  the  heavenly  bodies,  and  in  general,  in  all  investi- 
gations that  have  relation  to  their  apparent  positions  and  motions, 
Astronomers  conceive  all  these  bodies,  whatever  may  be  their 
actual  distance  from  the  earth,  to  be  referred  to  a  spherical  surface 
of  an  indefinitely  great  radius,  having  the  station  of  the  observer, 
or  what  comes  to  the  very  same  thing,  the  centre  of  the  earth, 
for  its  centre.  This  imaginaiy  spherical  surface  is  called  the 
Sphere  of  the  Heavens,  or,  the  Celestial  Sphere.  It  is  important 
to  observ^e,  that  by  reason  of  its  2:reat  dimensions,  if  two  lines 
be  draAvn  through  any  two  points  of  the  earth,  and  parallel  to 
each  other,  they  will,  when  prolonged  on,  meet  it  sensibly  in  the 
same  point ;  and  that,  if  two  parallel  planes  be  passed  through 
any  two  points  of  the  earth,  they  will  intersect  it  sensibly  in 
the  same  great  circle. 

26.  For  the  purposes  of  observation  and  computation,  certain 
imaginary  points,  lines,  and  circles,  appertaining  to  the  celestial 
sphere,  are  employed,  which  we  shall  now  proceed  to  explain. 

1.  The  Vertical  Line,  at  any  place  on  the  earth's  surface,  is 
the  line  of  descent  of  a  fallinor  body,  or  the  position  assumed  by 
a  plumb  line,  when  the  plummet  is  freely  suspended  and  at  rest. 

Every  plane  that  passes  through  the  vertical  line  is  called  a 
Vertical  Plane.  Eveiy  plane  that  is  perpendicular  to  the  vertical 
line,  is  called  a  Horizontal  Plane. 

2.  The  Sensible  Horizon  of  a  place  on  the  earth's  surface,  is 
the  circle  in  which  a  horizontal  plane,  drawn  through  the 
place,  cuts  the  celestial  sphere.  As  its  plane  is  tangent  to  the 
earth,  it  separates  the  visible  from  the  invisible  portion  of  the 
heavens. 

3.  The  Ratio7ial  Horizon  is  a  circle  parallel  to  the  former, 
the  plane  of  which  passes  tli  rough  the  centre  of  the  earth.  The 
zone  of  the  heavens  comprehended  between  the  sensible  and 


ON    THE    CELESTIAL    SPHERE.  9 

rational  horizon  is  imperceptible,  or  the  two  circles  appear  as 
one  and  the  same,  at  the  distance  of  the  earth. 

4.  The  Zenith  of  a  place  is  the  point  in  which  the  vertical 
prolono'ed  upwards  pierces  the  celestial  sphere.  The  point  in 
which  the  vertical,  when  produced  downwards,  intersects  the 
celestial  sphere,  is  called  the  Nadir. 

The  zenith  and  nadir  are  the  poles  of  the  horizon. 

5.  The  Axis  of  the  Heavens  is  an  imaginary  right  line,  pass- 
ing through  the  north  pole  (Art.  3)  and  the  centre  of  the  earth. 
It  is  the  line  about  which  the  apparent  rotation  of  the  heavens 
is  performed.  It  is,  also,  on  the  hypothesis  of  the  earth's 
rotation,  the  axis  of  rotation  of  the  earth  prolonged  on  to  the 
heavens. 

6.  The  South  Pole  of  the  heavens  is  the  point  in  which  the 
axis  of  the  heavens  meets  the  southern  part  of  the  celestial 
sphere. 

To  illustrate  the  preceding  definitions,  let  the  circle  N  E  S 
Q,  (Fig.  2.)  represent  the  earth,  and  O  Z  the  vertical  of  a  point 
O  on  its  surface.  Then,  H  O  R  will  be  the  plane  of  the  sensible 
horizon^  H'  C  R'  the  plane  of  the  rational  horizon,  O  Z  the  di- 
rection of  the  zenith,  and  O  C  that  of  the  ?iadir.  And,  if  O  P 
represent  the  direction  in  which  an  observer  at  O  will  see  the 
north  pole,  C  P,  parallel  to  O  P,  will  be  the  axis  of  the  heavens. 

Now,  neglecting  the  size  of  the  earth,  or  conceiving  the  ob- 
server to  be  stationed  at  its  centre,  let  C  (Fig.  3)  be  the  place  of 
observation,  C  Z  the  vertical  corresponding  to  O  Z  in  Fig.  2, 
and  P  the  north  pole  ;  then  will  Z  be  the  zenith  and  N  the 
nadir  ;  the  great  circle  H  A  R  a,  the  poles  of  which  are  Z,  N,  the 
horizon  ;  POP'  the  axis  of  the  heavens  ;  and  P'  the  sozdh  pole. 

7.  Vertical  Circles  are  great  circles  passing  through  the  ze- 
nith and  nadir.  They  cut  the  horizon  at  right  angles,  and  their 
planes  are  vertical.  Thus,  Z  S  M  represents  a  vertical  circle 
passing  through  the  star  S,  called  the  Vertical  Circle  of  the  Star. 

8.  The  Meridian  of  a  place  is  that  vertical  circle  which  con- 
tains the  north  and  south  poles  of  the  heavens.  The  plane  of 
the  meridian  is  called  the  Meridian  Plane. 

Thus,  P  Z  R  P'  is  the  meridian  of  the  station  C.     The  half  H 
Z  R,  above  the  horizon,  is  termed  the  Superior  Meridian,  and 
the  other  half  R  N  H,  below  the  horizon,  is  termed  the  Liferior 
2 


10  ASTRONOMY. 

Meridian.  The  two  points,  as  H  and  R,  in  which  the  meridian 
cuts  tlie  horizon,  are  called  the  North  and  South  Points  of 
the  horizon ;  and  the  line  of  intersection,  as  H  C  R,  of  the 
meridian  plane  with  the  plane  of  the  liorizon,  is  called  the  Me- 
ridian Line,  or  the  North  and  South  Line. 

9.  The  Prime  Vertical  is  the  vertical  circle  which  crosses  the 
meridian  at  right  angles.  It  cuts  the  horizon  in  two  points,  as 
e,  w,  called  the  East  and  West  Points  of  the  Horizon. 

10.  The  Altitude  of  any  heavenly  body  is  the  arc  of  a  verti- 
cal circle,  intercepted  between  the  centre  of  the  body  and  the 
horizon,  or  the  angle  at  the  centre  of  the  sphere,  measured  by 
this  arc.     Thus,  S  M  is  the  altitude  of  the  star  S. 

11.  The  Zenith  Distance  of  a  heavenly  body  is  the  arc  of  a 
vertical  circle,  intercepted  between  its  centre  and  the  zenith  ;  or 
the  distance  of  the  centre  of  the  body  from  the  zenith,  as  mea- 
sured by  the  arc  of  a  great  circle.  Thus,  Z  S  is  the  zenith  dis- 
tance of  the  star  S. 

It  is  obvious  that  the  zenith  distance  and  altitude  of  a  body 
are  compliments  of  each  other,  and,  therefore,  when  either  one 
is  known,  that  the  other  may  be  found. 

12.  The  Azimuth  of  a  heavenly  body  is  the  arc  of  the  hori- 
zon, intercepted  between  the  meridian  and  the  vertical  circle, 
passing  through  the  centre  of  the  body :  or  the  angle  compre- 
hended between  the  meridian  plane  and  the  vertical  plane  con- 
taining the  centre  of  the  body.  It  is  reckoned  either  from  the 
north  or  from  the  south  point,  and  each  way  from  the  meridian. 
H  M  represents  the  azimuth  of  the  star  S. 

The  Azimuth  and  Altitude,  or  azimuth  and  zenith  distance 
of  a  heavenly  body,  ascertain  its  position  with  respect  to  the 
horizon  and  meridian,  and,  therefore,  its  place  in  the  visible 
hemisphere.  Thus,  the  azimuth  H  M  determines  the  position  of 
the  vertical  circle  Z  S  M  of  the  star  S,  with  respect  to  the  meri- 
dian Z  P  H,  and  the  altitude  M  S,  or  the  zenith  distance  Z  S, 
the  position  of  the  star  in  this  circle. 

13.  The  Amplitude  of  a  heavenly  body  at  its  rising,  is  the 
arc  of  the  horizon  intercepted  between  the  point  where  the  body 
rises  and  the  east  point.  Its  amplitude  at  setting  is  the  arc  of 
the  horizon,  intercepted  between  the  point  where  the  body  sets 
and  the  west  point.    It  is  reckoned  towards  the  north,  or  towards 


ON    THE    CELESTIAL    SPHERE.  11 

the  south,  according  as  the  point  of  rising  or  setting  is  north  or 
south  of  the  east  or  west  point.  Thus,  if  a  B  S  A  represents 
the  circle  described  by  the  star  S  in  its  diurnal  motion,  e  a  will 
be  its  amplitude  at  rising,  and  w  A  its  amplitude  at  setting. 

14.  The  Celestial  Equator,  or  the  Equinoctial,  is  a  vertical  of 
the  celestial  sphere,  the  plane  of  which  is  perpendicular  to  the 
axis  of  the  heavens.  The  north  and  south  poles  of  the  heavens 
are  therefore  its  geometrical  poles.  The  celestial  equator  is  re- 
presented in  Fig.  3,  by  E  ■?/;  Q,  e.  This  circle  is  also  frequently 
called  the  Equator,  simply. 

15.  Parallels  of  Declination  are  small  circles,  parallel  to  the 
celestial  equator,  a  B  S  A  represents  the  parallel  of  declina- 
tion of  the  star  S. 

The  parallels  of  declination  passing  through  the  stars,  are 
the  circles  described  by  the  stars,  in  their  apparent  diurnal 
motion.  These,  by  way  of  abbreviation,  we  shall  call  Diurnal 
Circles. 

16.  Celestial  Meridians,  Hour  Circles,  and  Declination  Cir- 
cles, are  different  names  given  to  all  great  circles,  which  pass 
through  the  poles  of  the  heavens,  cutting  the  equator  at  right 
angles.  P  S  P'  is  a  celestial  meridian.  The  angles  compre- 
hended between  the  hour  circles  and  the  meridian,  reckoning 
from  the  meridian  towards  the  west,  are  called  Hour  Angles  or 
Horary  Angles. 

17.  The  Ecliptic  is  that  great  circle  of  the  heavens  which 
the  sun  appears  to  describe  in  the  course  of  the  year. 

18.  The  Obliquity  of  the  Ecliptic  is  the  angle  under  which 
the  ecliptic  is  inclined  to  the  equator. 

19.  The  Equinoctial  Points  are  the  tAvo  points  in  which 
the  ecliptic  intersects  the  equator.  That  one  of  these  points, 
which  the  sun  passes  in  the  spring,  is  called  the  Vernal 
Equinox,  and  the  other,  which  is  passed  in  the  autumn,  is 
called  the  Autumnal  Equinox.  These  terms  are  also  applied 
to  the  epochs  when  the  sun  is  at  the  one  or  the  other  of 
these  points. 

20.  The  Solstitial  Points  are  the  two  points  of  the  ecliptic 
90°  distant  from  the  vernal  and  autumnal  equinox.  The  one 
that  lies  to  the  north  of  the  equator  is  called  the  Summer  Sol- 
Miccj  and  the  other  the  Winter  Solstice.     The  epochs  of  the 


12  ASTRONOMY. 

sun's  arrival  at  these  points  are  also  designated  by  the  same 
terms. 

21.  The  Equinoctial  Colure  is  the  celestial  meridian  passing 
through  the  equinoctial  points ;  and  the  /Solstitial  Colure,  is 
the  celestial  meridian  passing  through  the  solstitial  points. 

22.  The  Polar  Circles  are  parallels  of  declination  at  a  dis- 
tance from  the  poles  equal  to  the  obliquity  of  the  ecliptic.  The 
one  about  the  north  pole  is  called  the  Arctic  Circle  ;  the  other, 
about  the  south  pole,  is  called  the  Antarctic  Circle. 

The  polar  circles  contain  the  poles  of  the  ecliptic. 

23.  The  Tropics  are  parallels  of  declination  at  a  distance 
from  the  equator  equal  to  the  obliquity  of  the  ecliptic.  That 
which  is  on  the  north  side  of  the  equator,  is  called  the  Tropic  of 
Cancer,  and  the  other,  the  Tropic  of  Capricorn. 

The  tropics  touch  the  ecliptic  at  the  solstitial  points. 

Let  C  (Fig.  4)  represent  the  centre  of  the  earth  and  sphere, 
C  P  the  axis  of  the  heavens,  E  V  Q.  A  the  equator,  W  V  L  A  the 
ecliptic,  and  K,  K',  its  poles.  Then  will  V  be  the  vernal  and  A 
the  autumnal  equinox ;  W  the  ivinter,  and  T  the  summer  sol- 
stice ;  P  V  P'  A  the  equinoctial  colure  ;  P  K  W  K'  T  the  solsti- 
tial colure  ;  the  angle  T  C  Q,  or  its  measure  the  arc  T  Q,,  the 
obliquity  of  the  ecliptic  ;  K  m,  U,  K'  7n'  U',  the  polar  circles  ; 
and  T  n  Z,  W  n'  Z',  the  tropics. 

24.  The  Zodiac  (Art.  13)  extends  about  9°  on  each  side  of 
the  ecliptic. 

25.  The  ecliptic  and  zodiac  are  divided  into  twelve  equal 
parts,  called  Signs.  Each  sign  contains  30°.  The  division 
commences  at  the  vernal  equinox.  Setting  out  from  this  point, 
and  following  around  from  west  to  east,  the  Signs  of  the  Zo- 
diac, with  the  respective  characters  by  which  they  are  desig- 
nated, are  as  follows  :  Aries  T,  Taurus  ^,  Gemini  n,  Cancer  o, 
Leo  SI,  Virofo  nj?,  Libra  d^,  Scorpio  m,  Sagittarius  /,  Capricor- 
nus  VJ,  Aquarius  'Zi,,  Pisces  X- 

A  motion  in  the  heavens  in  the  order  of  the  signs,  or  from 
west  to  east,  is  called  a  direct  motion,  and  a  motion  contrary 
to  the  order  of  the  signs,  or  from  east  to  west,  is  called  a 
retrograde  motion. 

26.  The  Right  Ascension  of  a  heavenly  body  is  the  arc  of  the 
equator,  intercepted  between  the  vernal  equinox  and  the  declina- 


ON    THE    CELESTIAL    SPHERE.  13 

tion  circle  which  passes  through  the  centre  of  the  body,  as 
reckoned  from  the  vernal  equinox  towards  the  east.  It  measures 
the  inclination  of  the  declination  circle  of  the  body  to  the  equi- 
noctial colure.  Thus,  P  S  R  being  the  declination  circle  of 
the  star  S,  and  V  the  place  of  the  vernal  equinox,  V  R  is  the 
right  ascension  of  the  star. 

27.  The  Declination  of  a  heavenly  body  is  the  arc  of  a  cir- 
cle of  declination,  intercepted  between  the  centre  of  the  body 
and  the  equator.  It  therefore  expresses  the  distance  of  the  body 
from  the  equator.     Thus,  R  S  is  the  declination  of  the  star  S. 

Declination  is  North,  or  South,  according  as  the  body  is 
north,  or  south  of  the  equator. 

The  right  ascension  and  declination  of  a  heavenly  body  are 
two  co-ordinates,  which,  taken  together,  fix  its  position  in  the 
sphere  of  the  heavens  :  for,  they  make  known  its  situation  with 
respect  to  two  circles,  the  equinoctial  colure,  and  the  equator. 
Thus,  V  R  and  R  S  ascertain  the  position  of  the  star  S  with  re- 
spect to  the  circles  P  V,  and  V  Q,  A  E. 

28.  The  Polar  Distance  of  a  heavenly  body  is  the  arc  of 
a  declination  circle,  intercepted  between  the  centre  of  the  body 
and  the  elevated  pole.  The  polar  distance  is  the  compliment 
of  the  declination,  and,  therefore,  when  either  is  known,  the 
other  may  be  found. 

29.  Circles  of  Latitude  are  great  circles  of  the  celestial 
sphere,  which  pass  through  the  poles  of  the  ecliptic,  and  there- 
fore cut  this  circle  at  right  angles.  Thus,  K  S  L  represents 
a  part  of  the  circle  of  latitude  of  the  star  S. 

30.  The  Longitude  of  a  heavenly  body  is  the  arc  of  the 
ecliptic,  intercepted  between  the  vernal  equinox  and  the  circle 
of  latitude,  which  passes  through  the  centre  of  the  body,  as 
reckoned  from  the  vernal  equinox  towards  the  east,  or  in  the 
order  of  the  signs.  It  measures  the  inclination  of  the  circle 
of  latitude  of  the  body  to  the  circle  of  latitude  passing 
through  the  vernal  equinox.  Thus,  V  L  is  the  longitude  of 
the  star  S. 

31.  The  Latitude  of  a  heavenly  body  is  the  arc  of  a  circle 
of  latitude,  intercepted  between  the  centre  of  the  body  and  the 
ecliptic.  It  therefore  expresses  the  distance  of  the  body  from 
the  ecliptic.     Thus,  L  S   is  the  latitude  of  the  star  S. 


14 


AS'I'HONO.M  \  . 


Latitude  is  Norths  or  iS'outh,  according  as  the  body  is  north,  or 
south  of  the  ecliptic. 

The  longitude  and  latitude  of  a  heavenly  body  are  another 
set  of  co-ordinates,  tohich  serve  to  fix  its  position  in  the  hea- 
vens. They  ascertain  its  situation  with  respect  to  the  circle 
of  latitude  passing  through  the  vernal  equinox  and  the  eclip- 
tic. Thus,  V  L  and  L  S  fix  the  position  of  the  star  S,  making 
known  its  situation  with  respect  to  the  circles  K  V  and  V  T 
AW. 

32.  The  Angle  of  Position  of  a  star  is  the  angle  included 
at  the  star  between  the  circles  of  latitude  and  declination 
passing  through  it.  P  S  K  is  the  angle  of  position  of  the 
star  i3. 

33.  The  Astro7iomical  Latitnde^  or  the  Latitvde  of  a  place, 
is  the  arc  of  the  meridian  intercepted  between  the  zenith  and 
the  celestial  equator.  It  is  North,  or  South,  according  as  the 
zenith  is  north,  or  south  of  the  equator.  Z  E  (Fig.  5)  re- 
presents the  latitude  of  the  station  o. 

27.  The  earth's  surface,  considered  as  spherical  (which  ac- 
curate admeasurement,  upon  principles  that  will  be  explained 
in  the  sequel,  proves  it  to  be,  very  nearly,)  is  called  the 
Terrestrial  Sjihere.  The  following  geometrical  constructions 
appertain  to  the  terrestrial  sphere,  as  it  is  employed  for  the 
purposes  of  astronomy.  It  will  be  observed  that  they  corre- 
spond to  those  of  the  celestial  sphere  above  described,  and 
are   used  for  similar   objects. 

1.  The  North  and  South  Poles  of  the  Earth  are  the  two 
points  in  which  the  axis  of  the  heavens  intersects  the  ter- 
restrial sphere.  They  are  also  the  extremities  of  the  earth's 
axis    of  rotation. 

2.  The  Terrestrial  Equator  is  the  great  circle,  in  which  a 
plane  passing  through  the  centre  of  the  earth,  and  perpendicular 
to  the  axis  of  the  heavens  and  earthy  cuts  the  terrestrial  sphere. 
The  terrestrial  and  the  celestial  equator,  then,  lie  in  the  same 
plane.  The  poles  of  the  earth  are  the  geometrical  poles  of  the 
terrestrial  equator.  The  two  hemispheres  into  which  the  ter- 
restrial equator  divides  the  earth,  are  called,  respectively,  the 
Northern  Hemisphere  and  the  Southern  Hemisphere. 

3.  Terrestrial  Meridians  are  great  circles  of  the  terrestrial 


ON    THE    TERRESTRIAL    SPHERE.  15 

sphere,  passing  through  the  north  and  south  poles  of  the  earth, 
and  cuttino;  the  equator  at  right  angles.  Every  plane  that  passes 
through  the  axis  of  the  heavens,  cuts  the  celestial  sphere  in  a 
celestial  meridian.,  and  the  terrestrial  sphere  in  a  terrestrial 
meridian. 

Let  P  P'  (Fig.  5,)  represent  the  axis  of  the  heavens,  O  the 
centre  of  the  earth,  and  p  and  p'  its  poles.  Then,  e  r  q  will 
represent  the  terrestrial  equator  (E  R  Q,  representing  the  celes- 
tial equator,)  and  pep'  and  p  s  p'  terrestrial  meridians  (P  E  P' 
and  P  S  P'  representing  celestial  meridians.) 

4.  The  Reduced  Latitude  of  a  place  on  the  earth's  surface  is 
the  arc  of  the  terrestrial  meridian,  intercepted  between  the  place 
and  the  equator,  or  the  angle  at  the  centre  of  the  earth  measured 
by  this  arc.  Thus,  o  e,  or  the  angle  o  O  e,  is  the  reduced  lati- 
tude of  the  place  o.  Latitude  is  North.,  or  South,  according  as 
the  place  is  north,  or  south  of  the  equator.  The  reduced  lati- 
tude differs  somewhat  from  the  astronomical  latitude,  by  reason 
of  the  slight  deviation  of  the  earth  from  a  spherical  form.  Their 
difference  is  called  the  Reduction  of  Latitude. 

5.  Parallels  of  Latitude  are  small  circles  of  the  terrestrial 
sphere  parallel  to  the  equator.  Every  point  of  a  parallel  of 
latitude  has  the  same  latitude. 

The  parallels  of  latitude  which  correspond  in  situation  with 
the  polar  circles  and  tropics  in  the  heavens,  have  received  the 
same  appellations  as  these  circles.     (See  Defs.  22,  23,  p.  12.) 

6  The  Longitude  of  a  place  on  the  earth's  surface,  is  the 
inclination  of  its  meridian  to  that  of  some  particular  station, 
fixed  upon  as  a  circle  to  reckon  from,  and  called  the  First  Meri- 
dian. It  is  measured  by  the  arc  of  the  equator,  intercepted 
between  the  first  meridian  and  the  meridian  passing  tlirouafh  the 
place,  and  is  called  East,  or  West,  according  as  the  latter  meri- 
dian is  to  the  east,  or  to  the  west  of  the  first  meridian.  Thus, 
if  ^  q  p'  be  supposed  to  represent  the  first  meridian,  the  angle 
s  p  q,  or  the  arc  q  I,  will  be  the  longitude  of  the  place  s. 

Different  nations  have,  for  the  most  part,  adopted  different 
first  meridians.  The  English  use  the  meridian  which  passes 
through  the  observatory  at  Greenwich,  near  London ;  and  th^^ 
French,  the  meridian  of  the  observatory  of  Paris.  In  the 
United  States,  as  we  have  no  public  observatory,  the  longitude 


16  ASTRONOMY. 

is,  for  astronomical  purposes,    reckoned  from  the  meridian  of 
Greenwich,  or  Paris,  (crenerally  the  former.) 

The  longitude  and  latitude  of  a  place  designate  its  situation 
on  the  eartKs  surface.  They  are  precisely  analogous  to  the 
right  ascension  and  declination  of  a  star  in  the  heavens. 

28.  The  diagram  (see  Fig.  3)  which  we  made  use  of  in 
Art.  26,  in  illustrating  our  description  of  the  circles  of  the 
celestial  sphere,  represents  the  aspect  of  this  sphere  at  a  place 
at  which  the  north  pole  of  the  heavens  is  somewhere  between 
the  zenith  and  horizon.  Such  is  the  position  of  the  north  pole 
at  all  places  situated  between  the  equator  and  the  north  pole  of 
the  earth.  For,  let  O  (Fis;.  6)  represent  a  place  on  the  earth's 
surface,  H  O  R  the  horizon,  O  Z  the  vertical,  H  Z  R  the  meri- 
dian, and  Z  E  the  latitude.  Q,  O  E  will  then  represent  the  equi- 
noctial, and  P,  P',  90°  distant  from  E  and  on  the  meridian,  the 
poles.     Now,  we  have, 

HP-ZH  —  ZP=90O  —  ZP;  ZE==PE  —  ZP  =  90o 
—  Z  P.     Whence  H  P  =  Z  E. 

Thus,  the  altitude  of  the  pole  is  every-where  equal  to  the 
latitude  of  the  place.  It  follows,  therefore,  that  in  proceeding 
from  the  equator  to  the  north  pole,  the  altitude  of  the  north 
pole  of  the  heavens  will  gradually  increase  from  0°  to  90°. 

If  the  spectator  is  in  the  southern  hemisphere,  the  elevated 
pole,  as  it  is  always  on  the  opposite  side  of  the  zenith,  from  the 
equator,  will  be  the  south  pole.  At  corresponding  situations  of 
the  spectator,  it  will  obviously  have  the  same  altitude  as  the 
north  pole  in  the  northern  hemisphere. 

29.  Let  us  now  inquire  into  the  principal  circumstances  of 
the  diurnal  motion  of  the  stars,  as  it  is  seen  by  a  spectator  situ- 
ated somewhere  between  the  equator  and  the  north  pole.  And, 
in  the  first  place,  it  is  a  simple  corollary  from  the  proposition 
just  established,  that  the  parallel  of  declination  to  the  north, 
whose  polar  distance  is  equal  to  the  latitude  of  the  place.,  will 
lie  entirely  above  the  horizon,  and  just  touch  it  at  the  north 
point.  This  circle  is  called  the  circle  of  perpetual  apparition  ; 
the  line  a  H  (Fig.  7)  represents  its  projection  on  the  meridian 
plane.  The  stars  comprehended  between  it  and  the  north  pole 
will  never  set.  As  the  depression  of  the  south  pole  is  equal  to 
the  altitude  of  the  north  pole,  the  parallel  of  declination  o  R, 


ASPECTS   OF    THE    CELESTIAL    SPHERE.  17 

at  a  distance  from  the  south  pole  equal  to  the  latitude  of  the 
place,  will  lie  entirely  below  the  horizon,  and  just  touch  it  at 
the  south  point.  The  parallel  thus  situated,  is  called  the  circle 
of  perj)etual  occultation.  The  stars  comprehended  betwen  it 
and  the  south  pole  will  never  rise. 

The  celestial  equator  (which  passes  through  the  east  and  west 
points)  will  intersect  the  meridian  at  a  point  E,  whose  zenith 
distance,  Z  E,  is  equal  to  the  latitude  of  the  place  (Def  33, 
Art.  26,)  and  consequently,  whose  altitude,  R  E,  is  equal  to 
the  co-latitude  of  the  j^lace.  Therefore,  in  the  situation  of  the 
observer  above  supposed,  the  equator  Q,  O  E,  passing  to  the 
south  of  the  zenith,  will,  together  with  the  diurnal  circles  n  r, 
s  t,  &c.,  which  are  all  parallel  to  it,  be  obliquely  inclined  to  the 
horizon,  making  with  it  an  angle  equal  to  the  co-latitude  of  the 
place.  As  the  centres  c,  c',  (fee,  of  the  diurnal  circles  lie  on  the 
axis  of  the  heavens,  which  is  inclined  to  the  horizon,  all  diurnal 
circles  situated  between  the  two  circles  of  perpetual  apparition 
and  occultation,  a  H  and  o  R,  with  the  exception  of  the  equator, 
will  be  divided  unequally  by  the  horizon.  The  greater  parts 
of  the  circles  n  r,  7i'  ?'',  <fcc.,  to  the  north  of  the  equator, 
will  be  above  the  horizon  ;  and  the  greater  parts  of  the  circles 
s  t,  s'  t\  (fee,  to  the  south  of  the  equator,  will  be  below'the  hori- 
zon. Therefore,  while  the  stars  situated  in  the  equator  will 
remain  an  equal  length  of  time  above  and  below  the  horizon, 
those  to  the  north  of  the  equator  will  remain  a  longer  time 
above  the  horizon  than  below  it ;  and  those  to  the  south  of  the 
equator,  on  the  contrary,  a  longer  time  below  the  horizon  than 
above  it.  It  is  also  obvious,  from  the  manner  in  which  the 
horizon  cuts  the  different  diurnal  circles,  that  the  disparity 
between  the  intervals  of  time  that  a  star  remains  above  and 
below  the  horizon,  will  be  the  greater  the  more  distant  it  is 
from  the  equator.  Again,  the  stars  will  all  culminate,  or  attain 
to  their  greatest  altitude,  in  the  meridian  :  for,  since  the  meridian 
crosses  the  diurnal  circles  at  right  angles,  they  will  have  the 
least  zenith  distance  when  in  this  circle.  Moreover,  as  the  meri- 
dian bisects  the  portions  of  the  diurnal  circles  which  lie  above 
the  horizon,  the  stars  will  all  employ  the  same  length  of  time  in 
passing  from  the  eastern  horizon  to  the  meridian,  as  in  passing 
from  the  meridian  to  the  western  horizon.  The  circumpolar 
3 


18  ASTRONOMY. 

stars  will  pass  the  meridian  tioice  in  24  hours  ;  once  above, 
and  once  below  the  pole.  These  meridian  passages  are  called 
respectively  Uppei'  and  Loioer  Culminations,  or  Inferior 
and  Superior  Transits. 

It  is  evident  from  what  is  stated  in  Art.  28,  that  the  circum- 
stances of  the  diurnal  motion  will  be  the  same  at  any  place  in 
the  southern  hemisphere,  as  at  the  place  which  has  the  same 
latitude  in  the  northern. 

The  celestial  sphere  in  the  position  relative  to  the  horizon 
which  we  have  now  been  considering,  which  obtains  at  all  places 
situated  between  the  equator  and  either  pole,  is  called  an  Oblique 
Sphere,  because  all  bodies  rise  and  set  obliquely  to  the  horizon. 

30.  When  the  spectator  is  situated  on  the  equator,  both  the 
celestial  poles  will  be  in  his  horizon,  (Art.  28,)  and,  therefore, 
the  celestial  equator  and  the  diurnal  circles  in  general  will  be 
perpendicular  to  the  horizon.  This  situation  of  the  sphere  is 
called  a  Right  Sphere,  for  the  reason  that  all  bodies  rise  and  set 
at  right  angles  with  the  horizon.  It  is  represented  in  Fig.  8. 
As  the  diurnal  circles  are  bisected  by  the  horizon,  the  stars 
will  all  remain  the  same  length  of  time  above,  and  below  the 
horizon. 

31.  If  the  observer  be  at  either  of  the  poles,  the  elevated  pole 
of  the  heavens  will  be  in  his  zenith,  (Art.  28,)  and,  consequently, 
the  celestial  equator  will  be  in  his  horizon.  The  stars  will  move 
in  circles  parallel  to  the  horizon,  and  the  whole  hemisphere,  on 
the  side  of  the  elevated  pole,  will  be  continually  visible,  while 
the  other  hemisphere  will  be  continually  invisible.  This  is 
called  a  Parallel  Sphere.     It  is  represented  in  Fig.  9. 


ASTRONOMICAL    INSTRUMENTS.  19 


CHAPTER    III. 

ON    THE    CONSTRUCTION    AND    USE    OF    THE    PRINCIPAL 
ASTRONOMICAL    INSTRUMENTS. 

32.  Astronomical  Instruments  are,  for  the  most  part,  used  for 
the  admeasurement  of  arcs  of  the  celestial  sphere,  or  of  angles 
corresponding  to  such  arcs  at  the  earth's  surface.  They  consist, 
essentially,  of  a  telescope  turning  upon  a  horizontal  axis,  and  of 
a  vertical  graduated  limb,  (or,  in  some  cases,  of  both  a  vertical 
and  a  horizontal  graduated  limb,)  to  indicate  the  angle  passed 
over  by  the  telescope.  At  the  common  focus  of  the  object-glass 
and  eye-glass  of  the  telescope  is  a  diaphragm,  or  circular  plate, 
attached  to  which  are  two  very  fine  wires,  or  threads,  crossing 
each  other  at  right  angles  in  its  centre.  The  place  of  this  dia- 
phragm may  be  altered  by  adjusting  screws  ;  it  is  by  this  means 
brought  into  such  a  position,  that  the  cross  of  the  wires  will  lie 
on  the  axis  of  the  telescope,  (that  is,  the  line  joining  the  centre  of 
the  object-glass  and  eye-glass.)  The  line  joining  the  centre  of 
the  object-glass  and  the  cross  of  the  wires,  is  technically  termed 
the  Line  of  Collimation.  Bringing  the  cross  of  the  wires  upon 
the  axis  of  the  telescope,  is  called  Adjttsting  the  Line  of  Colli- 
onation.  A  star  is  known  to  be  on  the  line  of  collimation  when 
it  is  bisected  by  the  cross- wires. 

The  telescope  either  turns  around  the  centre  of  the  graduated 
limb,  or,  which  is  more  common,  the  limb  and  telescope  are 
firmly  attached  to  each  other,  and  turn  together.  In  the  first 
arrangement,  a  small  steel  plate  firmly  connected  with  the  tele- 
scope, slides  along  the  limb.  Upon  this  plate  a  small  mark  is 
drawn,  which  is  called  the  Index.  The  required  angle  is  read 
^ffi  by  noting  the  angle  upon  the  limb  which  is  pointed  out  by 
the  index  ;  the  zero  on  the  limb  being  generally,  in  practice,  the 
point  from  which  the  angle  is  reckoned.  When  the  telescope 
and  graduated  limb  are  firmly  connected,  the  limb  slides  past 
the  index,  which  is  now  stationary.  The  limbs  of  even  the 
largest  instruments  are  not  divided  into  smaller  parts  than  about 


20  ASTRONOMY. 

5',  but,  by  means  of  certain  subsidiary  contrivances,  the  angle 
may,  with  some  instruments,  be  read  off  to  within  a  fraction  of 
a  second. 

33.  The  principal  contrivances  for  increasing  the  accuracy  of 
the  reading  off  of  angles,  are  the  Vernier,  the  Micrometer 
Screw,  and  the  Microscope  Micrometer.  The  Vernier  is  only 
the  index  plate,  so  graduated  that  a  certain  number  of  its  divi- 
sions occupy  the  same  space  as  a  number  one  less  on  the  limb. 
Fig.  10  represents  a  vernier,  and  a  portion  of  the  limb  of  the  in- 
strument, 15  divisions  on  the  vernier  corresponding  to  14  on  the 
limb.  If  we  suppose  the  smallest  divisions  of  the  limb  to  be  15', 
and  call  x  the  number  of  minutes  in  one  division  of  the  vernier, 

then, 

15  A'  =  14  X  15',  and  x  =  14'. 

Thus,  the  difference  between  a  division  on  the  vernier  and 
one  on  the  limb,  will  be  1'.  Accordingly,  if  the  index,  which  is 
the  first  mark  on  the  vernier,  should  be  little  past  the  mark  40° 
on  the  limb,  and  the  second  mark  of  the  vernier  should  coincide 
with  the  next  point  of  division,  marked  40°  15',  the  angle  would 
be  40°  1'.  If  the  third  mark  on  the  vernier  were  coincident 
with  the  next  division  of  the  limb,  the  angle  would  be  40°  2'. 
Kthe  fourth  with  the  next  division  to  this,  40°  3' ;  and  so  on. 

By  making  the  divisions  on  the  vernier  more  numerous,  the 
angle  can  be  read  off  with  greater  precision ;  but  a  better  expe- 
dient is  provided  m  the  Micrometer  Screw.  This  piece  of  me- 
chanism is  represented  in  Fig.  10.  The  part  E  can  be  fastened 
to  the  limb  of  the  instrument  by  means  of  a  screw.  F  G  is  a 
screw,  with  a  milled  head  at  F,  working  in  a  collar  fixed  m  the 
'under  part  of  E,  and  in  a  nut  fixed  in  the  under  part 
of  the  telescope  T  t.  When  the  part  E  is  fixed  or  clamped, 
and  the  screw  is  turned  around  by  its  milled  head  at  F,  it 
must  communicate  a  direct  motion  to  the  nut,  and,  conse- 
quently, to  the  telescope  and  vernier  in  the  direction  of  F  G. 
Attached  to  the  screw,  or  to  the  small  cylinder  on  which  it  is 
formed,  is  an  index  D,  moveable  together  with  the  screw,  and  on 
a  thin  graduated  immoveable  plate,  the  profile  only  of  Avhich  is 
sho\vn  in  the  figure.  Suppose  now  that  the  screw  is  of  such 
fineness,  that  while,  together  with  the  index  D,  it  makes  a  com- 
plete revolution,  the  vernier  moves  through  an  arc  of  1'.     Then, 


MICROSCOPE    MICROMETER — TIME,  &C.  21 

if  the  plate  be  divided  into  60  equal  parts,  a  motion  of  the  index 
over  one  of  these  parts  would  answer  to  a  motion  of  1"  on  the 
limb.  This  being  understood,  to  show  the  use  of  the  microme- 
ter screw,  suppose  that  no  two  marks  on  the  vernier  and  limb 
are  coincident :  bring  the  two  nearest  into  coincidence  by  turn- 
ing the  screw,  and  the  number  of  divisions  passed  over  by  the  in- 
dex D  will  be  the  seconds  to  be  added  to,  or  subtracted  from,  the 
angle  read  off  with  the  vernier.  In  observing  the  coincidence 
of  the  divisions  of  the  limb  and  vernier,  the  eye  is  assisted  by  a 
microscope. 

34.  The  Microscope  Micrometer  is  a  compound  microscope 
firmly  fixed  opposite  to  the  limb,  and  furnished  with  cross- 
wires  in  the  focus  of  the  eye-glass,  or  conjugate  focus  of  the 
object-glass,  moveable  by  a  fine  threaded  screw.  The  observer 
looks  through  it  at  the  limb.  By  turning  its  screw,  the  cross- 
wire  is  brought  into  exact  coincidence  with  the  nearest  of  the 
divisions  of  the  limb,  and,  as  with  the  micrometer  screw,  the 
distance  through  which  the  screw  is  turned  makes  known  the 
distance  from  this  division  of  the  fixed  centre  of  the  microscope, 
which  corresponds  to  the  index  of  a  fixed  vernier  plate. 

35.  It  is  obvious,  that,  other  things  being  the  same,  instru- 
ments are  accurate  in  proportion  to  the  power  of  the  telescope, 
and  the  size  of  the  limb.  The  large  instruments  now  in  use  in 
astronomical  observatories,  are  relied  upon  as  furnishing  angles 
to  within  1"  of  the  truth. 

36.  Time  is  an  essential  element  in  astronomical  observa- 
tion. Three  ditferent  kinds  of  time  are  employed  by  astrono- 
mers :  Sidereal,  Ajjparent  or  True  Solar,  and  Mean  Solar 
TiTTie, 

37.  Sidereal  TitJie  is  time  as  measured  by  the  diurnal  motion 
of  the  stars,  or,  more  properly,  of  the  vernal  equinox.  A  Side- 
real Day  is  the  interval  between  two  successive  meridian 
transits  of  a  star,  or,  (as  it  is  now  most  generally  considered,) 
the  interval  between  two  successive  transits  of  the  vernal 
equinox.  It  coimiiences  at  the  instant  when  the  vernal  equinox 
is  on  the  superior  meridian,  and  is  divided  into  24  Sidereal 
Hours. 

38.  Apparent,  or  True  Solar  Time,  is  deduced  from  obser- 
vations upon  the  sun.     An  Apparent  Solar  Dap  is  the  interval 


22  ASTRONOMY. 

between  two  successive  meridian  passages  of  the  sun's  centre ; 
commencing  Avhen  the  sun  is  on  the  superior  meridian.  It 
appears  from  observation,  that  it  is  a  little  longer  than  a  sidereal 
day,  and  that  its  length  is  variable  during  the  year.  It  is 
divided  into  24  Apparent  Solar  Hours. 

39.  Mean  Solar  Time  is  measured  by  the  diurnal  motion 
of  an  imaginary  sim,  called  tlie  Mean  Sun,  conceived  to  move 
uniformly  from  west  to  east  in  the  equator,  with  the  real  sun's 
mean  motion  in  the  ecliptic,  and  to  have  at  all  times  a  right 
ascension  equal  to  the  sun's  mean  longitude.  A  Mean  Solar 
Day  commences  when  the  mean  sun  is  on  the  superior  me- 
ridian, and  is  divided  into  24  Mean  Solar  Hours. 

Since  the  mean  sun  moves  uniformly  and  directly  towards 
the  east,  the  length  of  the  mean  solar  day  must  be  invariable. 

40.  The  Astronomical  Day  commences  at  noon,  and  is  di- 
vided into  24  hours ;  but  the  Calendar  Day  commences  at 
midnight,  and  is  divided  into  two  portions  of  12  hours  each. 

41.  Astronomical  observations  are,  for  the  most  part,  made 
in  the  plane  of  the  meridian.  But  some  of  minor  importance 
are  made  out  of  this  plane.  The  chief  instruments  employed 
for  meridian  observations,  are  the  Astronomical  Qnadranf,  and 
the  Transit  Instrument.,  in  connection  with  the  Astronomical 
Clock.  These  are  the  capital  instruments  of  an  observatory, 
inasmuch  as  they  serve  (as  will  soon  be  explained)  for  the  de- 
termination of  the  places  of  the  heavenly  bodies,  which  are  the 
fundamental  data  of  astronomical  science.  The  principal  in- 
struments used  for  observation  at  different  azimuths,  are  the 
Altitude  and  Azimuth  Instrument.,  the  Equatorial,  and  the 
Sextant. 

Transit  Instrument. 

42.  The  Transit  Instrument  is  a  meridional  instrument,  em- 
ployed in  conjunction  with  a  clock  or  chronometer  for  observing 
the  passage  of  celestial  objects  across  the  meridian,  either  for 
the  purpose  of  determining  their  difference  of  right  ascension, 
or  of  obtaining  the  correct  time.  Fig.  11  represents  a  fixed 
transit  instrument.  A  D  is  a  telescope,  fixed,  as  it  is  represented 
in  the  figure,  to  an  horizontal  axis  formed  of  two  cones.  The 
two  small  ends  of  these  cones  are  ground  into  two  perfectly 
equal   cylinders ;    which  cylindrical  ends   are  called   Pivots. 


TRANSIT    INSTllUMENT.  23 

These  pivots  rest  on  two  angular  bearings,  in  form  like  the 
upper  part  of  a  Y,  and  denominated  Y's.  The  Y's  are  placed 
in  two  dove-tailed  brass  grooves  fastened  in  two  stone  pillars 
E  and  W,  so  erected  as  to  be  perfectly  steady.  One  of  the 
grooves  is  horizontal,  the  other  vertical,  so  that,  by  means  of 
screws,  one  end  of  the  axis  may  be  pushed  a  little  forwards  or 
backwards,  and  the  other  end  may  be  either  slightly  depressed 
or  elevated :  which  two  small  movements  are  necessary,  as  it 
will  be  soon  explained,  for  two  adjustments  of  the  telescope. 

Let  E  be  called  the  eastern  pillar,  W  the  western.  On  the 
eastern  end  of  the  axis  is  fixed  (so  that  it  revolves  with  the  axis) 
an  index  w,  the  upper  part  of  which,  when  the  telescope  re- 
volves, nearly  slides  along  the  graduated  face  of  a  circle,  at- 
tached, as  it  is  shown  in  the  figure,  to  the  eastern  pillar.  The 
use  of  this  part  of  the  apparatus  is  to  adjust  the  telescope  to  the 
altitude  or  zenith  distance  of  a  star  the  transit  of  which  is  to  be 
observed.  Thus,  suppose  the  index  n  to  be  at  o,  in  the  upper 
part  of  the  circle,  when  the  telescope  is  horizontal :  then,  by 
elevating  the  telescope,  the  index  is  moved  downwards.  Sup- 
pose the  position  to  be  that  represented  in  the  figure,  then  the 
number  of  degrees  between  o  and  the  index  is  the  altitude. 

The  wire  plate  placed  in  the  focus  of  the  transit  telescope  has 
attached  to  it  five  vertical  wires,  together  with  one  horizontal 
wire.  In  order  to  be  seen  at  night,  these  wires  require  to  be 
illuminated  by  artificial  light.  Their  illumination  is  effected 
by  making  one  of  the  cones  hollow,  and  admitting  the  light  of 
a  lamp  placed  in  the  pillar  opposite  the  orifice  ;  which  light  is 
directed  to  the  wires  by  a  reflector  placed  diagonally  in  the 
telescope.  The  reflector,  having  a  large  hole  in  its  centre,  does 
not  interfere  with  the  rays  passing  down  the  telescope  from 
the  object. 

43.  We  will  now  explain  the  principal  adjustments  of  the 
transit.  Upon  setting  the  instrument  up,  it  should  be  so  placed, 
that  the  telescope,  when  turned  down  to  the  horizon,  should 
point  north  and  south,  as  near  as  can  possibly  be  ascertained. 
This  being  done,  then — 

1.  To  adjust  the  litie  of  collimation. 

This  adjustment  consists  in  bringing  the  central  vertical  wire, 
within  the  telescope,  to  intersect  the  optical  axis,  which  is  sup- 


24  ASTRONOMY. 

posed  to  be  fixed  by  the  maker  of  the  instrument  perpendicu- 
larly to  the  axis  of  rotation.  There  is  no  occasion  with  this 
instrument,  to  have  the  horizontal  wire  intersect  the  optical 
axis  with  exactness.  Direct  the  telescope  to  some  small,  dis- 
tant, well  defined  object,  (the  more  distant  the  better,)  and  bisect 
it  with  the  middle  of  the  central  vertical  wire ;  then  lift  the 
telescope  out  of  its  angular  bearings,  or  Y's,  and  replace  it,  with 
the  axis  reversed.  Point  the  telescope  again  to  the  same  object, 
and  if  it  be  still  bisected,  the  collimation  adjustment  is  correct ; 
if  not,  move  the  wires  one  half  the  angle  of  deviation,  by  turn- 
ing the  small  screws  that  hold  the  wire  plate,  near  the  eye  end 
of  the  telescope,  and  the  adjustment  will  be  accomplished  ;  but, 
as  half  the  deviation  may  not  be  correctly  estimated  in  moving 
the  wires,  it  becomes  necessary  to  verify  the  adjustment,  by 
moving  the  telescope  the  other  half,  which  is  done  by  turning 
the  screw  that  gives  the  small  azimuth  motion  to  the  Y  before 
spoken  of,  and  consequently,  to  the  pivot  of  the  axis  which  it 
carries.  Having  thus  again  bisected  the  object,  reverse  the  axis 
as  before,  and  if  half  the  error  was  correctly  estimated,  the 
object  will  be  bisected  upon  the  telescope  being  directed  to  it. 
If  it  should  not  be  bisected,  the  operation  of  reversing  and  cor- 
rectinof  half  the  error  must  be  orone  throuarh  ag'ain,  and,  until  af- 
ter  successive  approximations  the  object  is  found  to  be  bisected 
in  hvlh  positions  of  the  axis  ;  the  adjustment  will  then  be 
perfect. 

It  is  desirable  that  the  central  wire  should  be  truly  vertical, 
as  we  should  then  have  the  power  of  observing  the  transit  of  a 
star  on  any  part  of  it,  as  well  as  the  centre.  It  may  be  ascer- 
tained whether  it  is  so,  by  elevating  and  depressing  the  telescope, 
when  directed  to  a  distant  object ;  if  it  is  bisected  by  every  part 
of  the  wire,  the  wire  is  vertical ;  if  it  is  not  bisected,  the  wire 
should  be  adjusted,  by  turning  the  inner  tube  carrying  the  wire 
plate  until  the  above  test  of  its  verticality  be  obtained. 

2.    To  set  the  axis  of  rotation  of  the  telescope  horizontal. 

44.  This  adjustment  may  be  effected  by  means  of  a  spirit-level, 
attached  to  two  upright  arms  bent  at  their  upper  extremities,  by 
which  it  is  hung  on  the  two  pivots  of  the  axis.  At  one  end  of 
the  level  is  a  vertical  adjusting  screw,  by  which  that  end  may 
be  elevated  or  depressed.     Put  the  telescope  in  its  place,  and 


TRANSIT    INSTRUMENT.  25 

observe  to  which  end  of  the  level  the  bubble  runs,  which  will 
always  be  the  more  elevated  end  ;  bring  it  back  to  the  middle 
by  the  Y  screw  for  vertical  motion,  and  take  off  the  level,  and 
hang  it  on  again  with  the  ends  reversed.  Then,  if  the  bubble  is 
again  found  in  the  middle,  the  level  is  already  parallel  to  the 
axis,  and  the  axis  horizontal  ;  but  if  not,  adjust  one  half  the 
error  by  the  adjusting  screw  of  the  level,  and  the  other  half  by 
the  Y  screw  ;  and  let  the  operation  of  reversing,  and  adjusting 
by  halves,  be  repeated  until  the  bubble  will  remain  stationary 
in  either  position  of  the  level,  in  which  case,  both  the  level  and 
axis  will  be  horizontal. 

3.  To  adjust  tlie  line  of  collimation  to  the  plane  of  the 
meridian. 

45.  We  have  said,  that  upon  setting  the  instrument  up,  the 
telescope  is  to  be  brought  into  the  meridian  plane,  as  near  as 
can  be  ascertained.  One  mode  of  establishing  it,  is  to  direct  the 
telescope  to  the  pole  star,  and  by  repeated  observations  find  the 
position  corresponding  to  its  greatest  or  least  altitude.  At  the 
present  time,  we  may,  instead,  compute  by  means  of  existing 
tables  founded  on  observ^ation,  the  time  of  the  meridian  transit 
of  the  pole  star,  and  at  that  computed  time  bisect  the  star  by 
the  middle  vertical  wire.  Afterwards,  the  line  of  collimation 
may  be  placed  still  more  exactly  in  the  plane  of  the  meridian,  in 
the  following  manner  :  Note  the  times  of  two  successive  supe- 
rior transits  of  the  pole  star  across  the  central  vertical  wire,  and 
the  time  of  the  intervening  inferior  transit.  If  the  line  of  colli- 
mation were  exactly  in  the  plane  of  the  meridian,  as  the  diurnal 
circles  are  bisected  by  this  plane,  the  interval  between  the  supe- 
rior and  next  inferior  transit,  would  be  precisely  equal  to  the 
interval  between  the  inferior  and  next  superior  transit.  Accord- 
ingly, if  these  intervals  arc  not  in  fact  equal,  find  by  repeated 
trials,  the  position  of  the  telescope  and  vertical  wire  for  which 
they  are  equal,  and  the  line  of  collimation  will  then  be  in  the 
plane  of  the  meridian.  The  method  of  regulating  the  clock, 
required  in  making  this  adjustment,  will  be  explained  when  we 
come  to  treat  of  the  astronomical  clock. 

46.  When  the  transit  telescope  has  once  been  placed  accu- 
rately in  the  meridian  plane,  in  order  to  avoid  the  repetition  of 
troublesome  verifications  of  its  position,  a  meridian  mark  should 

4 


26  ASTRONOMY. 

be  set  up,  and  permanently  established,  at  a  distance  from  the 
instrument :  its  place  being  determined  by  means  of  the  middle 
or  meridional  wire.  At  Greenwich,  two  such  marks,  one  to  the 
nortli  and  another  to  the  south,  are  used  ;  they  are  vertical 
stripes  of  white  paint  upon  a  black  ground,  on  buildings  about 
two  miles  distant  from  the  observ^atory.  The  position  of  the 
telescope  is  verified,  by  sighting  at  the  meridian  mark,  when  it 
is  once  established. 

47.  The  times  of  the  transits  of  the  heavenly  bodies  are 
ascertained  as  follows :  In  the  case  of  a  star,  the  moments  of 
its  crossing  each  of  the  five  vertical  wires  are  noted ;  as  the 
wires  are  equally  distant  from  each  other,  the  mean  of  these 
times  (or  their  sum  divided  by  5)  will  be  the  time  of  the  stars 
crossing  the  middle  wire,  or  of  its  meridian  transit.  The  utility 
of  having  five  wires,  instead  of  the  central  one  only,  will  be 
readily  understood,  from  the  consideration  that  a  mean  result 
of  several  observations  is  deserving  of  more  confidence  than  a 
single  one  ;  since  the  chances  are,  that  an  error  which  may  have 
been  made  at  one  wire,  will  be  compensated  by  an  opposite  error 
at  an  other.  If  the  body  observed  has  a  disc  of  perceptible  mag- 
nitude, as  in  the  cases  of  the  sun,  moon,  and  planets,  the  times 
of  the  passage  of  both  the  western  and  eastern  limb  across  each 
of  the  five  wires  are  noted,  and  the  mean  of  the  whole  taken, 
which  will  be  the  instant  of  the  meridian  transit  of  the  centre  of 
the  body. 

The  time  of  the  transit  of  a  body,  may,  in  this  manner,  be 
ascertained  within  a  few  tenths  of  a  second. 

48.  In  the  interval  between  the  transits  of  any  two  stars,  the 
arc  of  the  equator  which  expresses  their  difference  of  right 
ascension,  will  pass  across  the  meridian,  the  rate  of  the  motion 
being  that  of  15°  to  a  sidereal  hour:  hence  the  difference  of 
the  times  of  transit  of  two  stars,  as  observed  with  a  sidereal 
clock,  when  converted  into  degrees,  by  allowing  15°  to  the 
hour,  will  be  the  difference  between  the  right  ascensions  of  the 
two  stars.  We  may  then,  in  this  manner,  by  means  of  a  transit 
instrument  and  sidereal  clock,  find  the  differences  between  the 
right  ascension  of  any  one  star  and  the  right  ascensions  of  all 
the  others.  This  being  done,  as  soon  as  the  position  of  the  ver- 
nal equinox  with  respect  to  the  same  star  becomes  known,  (and 


ASTRONOMICAL    CLOCK.  27 

we  shall  show  how  to  find  it  in  the  sequel,)  the  absolute  right 
ascensions  of  all  the  stars  will  also  become  known.  In  the  ac- 
tually existing-  state  of  astronomical  science,  the  right  ascensions 
of  all  the  stars  are  more  or  less  accurately  known,  and  a  right 
ascension  sought  is  now  obtained  directly,  by  noting  the  time  of 
the  transit  of  the  body,  with  a  sidereal  clock  resrulated  so  as  to 
indicate  Oh.  Om.  Os.,  when  the  vernal  equinox  is  on  the  meri- 
dian, and  converting  it  into  degrees. 

Astronomical  Clock. 

49.  The  astronomical  clock  is  very  similar  to  the  common 
clock.  It  has  a  compensation  pendulum ;  that  is,  a  pendulum 
so  constructed,  that  its  leno-th  is  unaffected  by  changes  of  tem- 
perature.    The  hours  on  the  face  are  marked  from  1  to  24. 

50.  Astrono  iiers  make  use  of  sidereal  time  (as  already  stated) 
in  determining  the  right  ascensions  of  the  heavenly  bodies,  but 
for  all  other  purposes,  they  generally  use  mean  solar  time. 

51.  To  regulate  a  sidereal  clock.  When  a  clock  is  used 
for  determining  differences  of  right  ascension,  (Art.  48,)  it  is 
adjusted  to  sidereal  time,  if  it  goes  equally  and  marks  out 
24  hours  in  a  sidereal  day  ;  it  being  altogether  immaterial 
at  what  time  it  indicates  Oh.  Om.  Os.  To  ascertain  the  daili/ 
rate  of  going  of  a  clock  which  is  to  be  adjusted  to  sidereal 
time  for  the  purpose  just  mentioned,  note  by  the  clock  the 
times  of  two  successive  meridian  transits  of  the  same  star. 
The  difference  between  the  interval  of  the  transits  and  24 
hours,  will  be  the  daily  gain  or  loss  (as  the  case  may  be)  of 
the  clock  with  respect  to  a  perfectly  accurate  sidereal  clock.* 
If  the  gain  or  loss,  when  found  after  this  manner,  proves  to  be 
the  same  each  day,  then  the  mean  rate  of  going  is  the  same 
each  day. 

Next,  to  be  able  to  discover  the  rate  from  hour  to  hour  during 
the  day,  it  is  necessary  to  have  obtained  beforehand,  at  various 
times,  and  under  various  states  of  the  circumstances  likely  to 
influence  the  rate  of  going  of  the  clock,  the  differences  between 
the  times  of  the  transits  of  a  number  of  different  stars,  (correct- 

*  It  is  not  necessary,  in  order  to  obtain  the  daily  rate  of  a  sidereal  clock,  that 
the  transit  instrument  should  be  adjusted  to  the  plane  of  the  meridian.  It  is  only 
requisite  that  it  should  be  kept  fixed  in  some  one  vertical  plane. 


28  ASTRONOMY. 

ing  proportionally  for  the  daily  rate,)  and  to  take  the  mean  of 
the  several  differences  found  for  each  pair  of  stars  for  the  exact 
difference  of  their  transits.  When  this  has  been  done,  the  rate 
of  the  clock  may  be  found  at  all  hours  during  the  day,  by  noting 
by  the  clock  tJie  differences  between  the  times  of  the  transits 
of  these  stars,  and  comparing  these  with  the  exact  differences 
already  found.  At  the  present  time,  the  right  ascensions  of  the 
stars  being  known,  we  have  only,  in  order  to  ascertain  the  rate 
from  hour  to  hour,  to  compare  the  intervals  of  time  given  by 
the  clock  between  the  transits  of  different  stars  taken  in  the 
order  of  their  right  ascension,  with  their  differences  of  right 
ascension. 

52.  The  sidereal  clocks  now  in  use,  are  made  to  indicate 
Oh.  Om.  Os.  when  the  vernal  equinox  is  on  the  superior  meri- 
dian. For  the  regulation  of  such  clocks,  it  is  necessary  to 
know  not  only  their  rate,  but  also  their  error.  This  is  found 
by  noting  the  time  of  the  trauoit  of  a  star,  and  comparing  it 
with  its  right  ascension  expressed  in  time.  If  the  two  are  equal, 
the  clock  is  right,  otherwise  their  difference  will  be  its  error. 

If  the  error  of  the  rate  of  a  clock  be  considerable,  it  should 
be  diminished  by  altering  the  length  of  the  pendulum  ;  other- 
wise, it  may  be  allowed  for.  The  stars  best  adapted  to  the 
regulation  of  clocks,  are  those  in  the  vicinity  of  the  equator ; 
for,  as  their  motion  is  more  rapid  than  that  of  the  stars  more 
distant  from  the  equator,  there  is  less  liability  to  error  in  noting 
their  transits, 

53.  A  mean  solar  clock  is  usually  regulated  by  observations 
upon  the  sun.  The  method  of  regulating  it  cannot  be  ade- 
quately explained  until  we  have  treated  of  the  apparent  motions 
of  the  sun.  It  will  here  suffice  to  state,  that  with  the  instru- 
ments we  have  now  described  the  sun's  motions  can  be  ascer- 
tained ;  and  that,  therefore,  as  a  knowledge  of  these  is  all  that 
is  necessary,  in  order  that  we  may  be  able  to  obtain  the  mean 
solar  time  at  any  instant,  it  is  ])ossible  to  express  all  intervals 
of  time  in  mean  solar  time. 

Astronomical   Quadrant. 

54.  An  Astronomical  Quadrant  is  an  instrument  designed  for 
the  measurement  of  the  meridian  zenith  distances  or  altitudes 
of  the  heavenly  bodies,  of  which  the  essential  parts  are  a  grad- 


ASTRONOMICAL    QUADRANT.  29 

uated  quadrantal  limb,  a  telescope  turning  upon  an  axis  which 
passes  through  the  centre  of  the  limb,  and  a  plumb  line  or 
level,  to  ascertain  the  vertical  or  horizontal  line.  (See  Fig.  12.) 
Astronomical  quadrants  may  be  either  portable  or  fixed.  Port- 
able quadrants  are  mounted  upon  an  upright  stem  resting  upon 
a  tripod,  and  can  be  turned  around  in  azimuth.  Fixed  quad- 
rants have  their  axis  firmly  fastened  in  a  wall,  in  a  horizontal 
position,  and  perpendicularly  to  the  plane  of  the  meridian. 
They  are  hence  called  Mural  Quadrants.  The  large  mural 
quadrants  of  the  Greenwich  Observatory  are  of  8  feet  radius. 

55.  The  same  adjustments  are  necessary  for  the  quadrant  as 
for  the  transit  instrument ;  and  in  addition,  the  horizontal  wire 
must  be  brought  to  intersect  the  axis,  and  the  vertical,  or  hori- 
zontal point  on  the  limb,  must  be  found.  The  methods  of  ef- 
fecting the  adjustments  are  also  the  same  with  the  fixed  quad- 
rant, except  in  the  case  of  the  collimation  adjustment.  This 
cannot  be  effected  without  the  intervention  of  a  portable  quad- 
rant or  similar  instrument.  The  horizontal  wire  of  the  tele- 
scope of  a  portable  quadrant  may  be  brought  to  intersect  the 
optical  axis,  by  directing  the  telescope  to  some  star  near  the 
zenith,  bisecting  it  with  the  horizontal  wire,  then  turning  the 
instrument  180°  in  azimuth,  and  moving  the  wire  over  half  its 
angular  distance  from  the  star  in  the  new  position  of  the  tele- 
scope, and  repeating  the  process  until  the  star  is  bisected  in  both 
positions  of  the  instrument.  This  adjustment  may  be  avoided 
altogether,  by  taking  the  half  difference  of  the  zenith  distances 
of  the  star  in  the  two  positions  of  the  instrument  for  the  con- 
stant index  error.  The  horizontal  point  (technically  so  called) 
is  the  place  of  the  index  answering  to  the  horizontal  position 
of  the  telescope.  It  may  be  found  by  means  of  a  level  or 
plumb  line. 

56.  In  place  of  mural  quadrants,  Mural  Circles  are  often 
used.  For  greater  accuracy,  the  angle  is  read  off  at  six  differ- 
ent points  of  the  limb,  which  is  an  entire  circle,  by  means  of 
six  stationary  microscope  micrometers,  (Art.  34,)  and  the  mean 
of  the  different  readings  taken  for  the  angle  required. 

57.  There  is  another  modification  of  the  quadrant,  called  the 
Zenith  Sector,  which  is  used  to  measure  the  meridian  zenith 
distances  of  stars  that  cross  the  meridian  within  a  few  degrees 


30 


ASTRONOMY. 


of  the  zenith.  The  limb  extends  only  about  10*^  on  each  side 
of  the  lowest  point.  The  zenith  sector  in  the  observatory  at 
Greenwich  has  a  radius  of  12  feet. 

58.  The  meridian  altitude  of  a  star  is  obtained  by  bringing 
the  telescope  into  such  a  position,  that  the  star  will  be  bisected 
by  the  horizontal  wire  as  it  passes  through  the  field  of  view,  and 
observing  the  angle  upon  the  limb.  That  of  the  sun,  moon,  or 
any  planet,  may  be  ascertained  by  measuring  the  altitudes  of 
the  upper  and  lower  limbs,  and  taking  their  half  sum  for  the 
altitude  of  the  centre :  or,  if  the  apparent  semidiameter  be 
known,  by  adding  this  to  the  altitude'  of  the  lower  limb,  or 
subtracting-  it  from  the  altitude  of  the  upper  limb. 

59.  The  meridi  ,n  altitude  or  zenith  distance  of  a  heavenly 
body  having  been  measured  with  an  astronomical  quadrant,  or 
other  similar  instrument,  at  a  place  the  latitude  of  which  is 
known,  its  declination  may  easily  be  found.  For,  let  s,  s',  s" 
(Fig.  6,)  represent  the  points  of  meridian  passage  of  three  differ- 
ent stars ;  one  to  the  north  of  the  zenith  (Z,)  one  between  the 
zenith  and  the  equator  (E,)  and  a  third  to  the  south  of  the 
equator,  and  we  sliall  have— 

E  5  =  Z  E  +  Z  5,  E  5'  =  Z  E  —  Z  5',  E  5"  =  Z  s"  —  Z  E  = 

— (ZE  — Z5"); 

or,  in  general. 

Declination  --latitude  +  zenith  distance  .  .  .  (1)  ; 
the  latitude  being  taken  always  positive,  the  zenith  distance 
being  also  taken  positive  when  of  the  same  name  with  the  lati- 
tude, but  negative  when  of  a  contrary  name  ;  and  the  declina- 
tion being  north,  if  it  comes  out  positive,  and  south,  if  it  comes 
out  negative. 

The  latitude,  which  is  here  supposed  to  be  known,  may  be 
found  by  measuring  the  meridian  altitudes  of  a  circumpolar 
star  at  its  inferior  and  superior  transits,  and  taking  their  half 
sum.  For,  as  the  pole  lies  midway  between  the  points  at  which 
the  transits  take  place,  its  altitude  will  be  the  arithmetical  mean, 
or  the  half  sum  of  the  altitudes  of  these  points,  and  the  altitude 
of  the  pole  is  equal  to  the  latitude  of  the  place  (Art.  28.) 

60.  When  the  right  ascension  and  declination  of  a  heavenly 
body  have  been  obtained  from  observation,  with  a  transit  instru- 
ment and  quadrant,  (Arts.  48,  59,)  its  longitude  and  latitude  may 


ALTITUDE    AMD    AZIMUTH    INSTRUMENT — EQUATORIAL.      31 

be  computed.  For,  Let  S  (Fig.  4)  represent  the  place  of  the 
body,  V  R  Gl  E  the  equator,  V  L  T  W  the  ecliptic,  and  P,  K,  the 
poles  of  the  equator  and  ecliptic.  In  the  triangle  P  K  S  we 
shall  know,  P  S  the  compliment  of  S  R  the  declination,  and  the 
angle  K  P  S  =  E  R  ^  E  V  +  V  R  =  90°  +  right  ascension  ;  and, 
if  we  suppose  the  obliquity  of  the  ecliptic  to  be  known,  we  shall 
know  P  K.  We  may  therefore  compute  K  S,  and  the  angle  P 
K  S.  But  K  S  is  the  compliment  of  S  L,  which  is  the  latitude 
of  the  body  S  ;  and  P  K  S  180°  —  E  K  S  ^  180°  —  (W  V  + 
V  L)  =  180°  —  (90°  +  longitude)  =  90°  —  longitude. 

The  obliquity  of  the  ecliptic,  which  we  have  here  supposed  to 
be  known,  is,  in  practice,  easily  found  ;  for.  it  is  equal  to  S  Q,  the 
sun's  greatest  declination. 

Altitude  and  Azimuth  Instrument. 

61.  The  Altitude  and  Azimuth  Instrument  consists,  essentially, 
of  a  telescope  with  two  graduated  limbs,  the  one  horizontal,  and 
the  other  vertical.  The  telescope  turns  about  the  centre  of  the 
vertical  limb,  or  turns  with  the  limb  about  its  centre;  and  the 
vertical  limb  turns,  with  the  telescope,  about  the  vertical  axis  of 
the  ho:-  zontal  limb. 

If  the  telescope  be  brought  into  the  meridian  plane,  and  after- 
wards directed  upon  a  star  out  of  this  plane,  the  arc  of  the  hori- 
zontal limb  passed  over  by  the  index  will  be  the  azimuth  of  the 
star.     The  vertical  limb  will  serve  to  measure  its  altitude. 

62.  The  Meridian  Line  at  a  place  may  easily  be  determined 
with  the  altitude  and  azimuth  instrument,  by  a  method  called 
the  Method,  of  Equal  Altitudes.  Let.  O  (Fig.  13)  represent  the 
place  of  observation,  N  P  Z  the  meridian,  and  S,  S'  two  posi- 
tions of  the  same  star,  at  which  the  altitude  is  the  same.  Now, 
the  triangles  Z  P  S  and  Z  P  S'  have  the  side  Z  P  common,  Z 
S  =  Z  S',  and  (allowing  that  the  stars  move  in  circles)  PS  P 
S'.  Hence,  they  are  equal,  and  consequently  the  angle  P  Z  S  -  P 
Z  S' ;  that  is,  equal  altitudes  of  a  star  correspond  to  equal  azi- 
omUhs.  Therefore,  by  bisecting  the  arc  of  the  horizontal  limb, 
comprehended  between  two  positions  of  the  vertical  limb  for 
which  the  observed  altitude  of  a  star  is  the  same,  we  shall  obtain 
the  meridian  line. 

Equatorial. 

63.  The  Equatorial  is  very  similar,  in  its  construction,  to  the 


32  ASTRONOMY. 

altitude  and  azimuth  instrument.  It  is  so  called,  from  the  cir- 
cumstance of  one  of  the  limbs  being  placed  in  a  position  parallel 
to  the  plane  of  the  equator.  The  axis  of  this  limb  is  then  pa- 
rallel to  the  axis  of  the  heavens  ;  and  the  limb,  to  the  centre  of 
which  the  telescope  is  attached,  is  parallel  in  every  one  of  its 
positions  to  tlie  plane  of  some  one  celestial  meridian.  This  in- 
strument is  particularly  useful  in  the  measurement  of  apparent 
diameters,  and  in  all  observations  that  require  the  telescope  to 
be  directed  upon  a  body  for  a  considerable  period  of  time  ;  as, 
by  givino;  the  limb  to  which  the  telescope  is  attached  a  slow 
motion  from  east  to  west,  the  body  may  be  followed  in  its  diur- 
nal motion,  and  kept  continually  within  the  field  of  view. 

Sexta7it. 
64.  The  Sextant  serves  for  the  direct  admeasurement  of  the 
angular  distance  between  any  two  objects.  Its  essential  parts 
are  a  graduated  limb  B  C  (Fig.  14),  comprising  about  60  de- 
grees of  the  entire  circle,  which  is  attached  to  a  triangular 
frame  BAG;  two  mirrors,  of  which  one  (A),  called  the  Index 
Glass,  is  moveable  in  connection  with  an  index  G  about  A,  the 
centre  of  the  limb,  and  the  other  (D),  called  the  Horizon- Glass , 
is  permanently  fixed  parallel  to  the  radius  A  C  drawn  to  the 
zero  point  of  the  limb,  and  is  only  half-silvered,  (the  upper  half 
being  transparent ;)  and  an  immoveable  telescope  at  E,  directed 
towards  the  horizon-glass.  The  principle  of  the  construction 
and  use  of  the  sextant  may  be  understood  from  what  follows. 
A  ray  of  light  S  A  from  a  celestial  object  S,  which  impinges 
against  the  index-glass,  is  reflected  off"  at  an  equal  angle,  and 
striking  the  horizon-glass  (D),  is  again  reflected  to  E,  where  the 
eye  likewise  receives  through  the  transparent  part  of  that  glass, 
a  direct  ray  from  another  point  or  object  S'.  Now,  if  A  S'  be 
drawn,  directed  to  the  object  S',  S  A  S',  the  angular  distance  be- 
tween the  two  objects  S  and  S'  is  equal  to  double  the  angle  C  A 
G,  measured  upon  the  limb  of  the  instrument  (A  C  being  parallel 
to  the  horizon -glass.)  For,  when  the  index-glass  is  parallel  to 
the  horizon-glass  and  the  angle  on  the  limb  is  zero,  A  D,  the 
course  of  the  first  reflected  ray,  will  make  equal  angles  with  the 
two  glasses,  and  therefore  the  angle  SAD  will  become  the  angle 
S'AD(=AD  E) ;  and  the  observer,  looking  through  the  tele- 
scope, will  see  the  same  object  S'  both  by  direct  and  reflected 


SEXTANT.  33 

lio-ht.  Now,  if  the  index-glass  be  moved  from  this  position 
through  any  angle  C  A  G,  the  angle  made  by  the  reflected 
ray  A  D  with  this  glass,  will  be  diminished  by  an  amount  equal 
to  this  angle  ;  for,  we  have  DAG=  DAC  —  CAG.  There- 
fore, the  angle  made  by  the  incident  ray  witli  the  index-glass, 
as  it  is  always  equal  to  that  made  by  the  reflected  ray,  will  be 
diminished  by  this  amount.  Consequently,  the  incident  ray  will, 
on  the  whole,  that  is,  by  the  diminution  of  its  inclination  to 
the  mirror  by  the  angle  CAG,  and  by  the  motion  of  the  mirror 
through  the  same  angle,  be  displaced  towards  the  right  an  angle 
S  A  S',  equal  to  2  G  A  C.  Thus,  the  angular  distance  S  A  S' 
of  two  objects  S,  S',  seen  in  contact,  the  one  (S')  directly,  and 
the  other  (S)  by  reflection  from  the  two  mirrors,  is  equal  to  twice 
the  angle  CAG  that  the  index-ghiss  is  moved  from  the  position 
(A  C)  of  parallelism  to  the  horizon-glass. 

Hence  the  limb  is  divided  into  120  equal  parts,  which  are 
called  desrrees ;  and  to  obtain  the  anovular  distance  between 
two  points,  it  is  only  necessary  to  sight  directly  at  one  of  them, 
and  then  move  the  index  until  the  reflected  image  of  the  other 
is  brought  into  contact  with  it ;  the  angle  read  off"  on  the  limb 
will  be  the  anorle  souofht. 

65.  To  obtain  the  angular  distance  between  two  bodies 
which  have  a  sensible  diameter,  bring  the  nearest  limbs  into 
contact,  and  to  the  angle  read  off"  on  the  limb  add  the  sum  of 
the  apparent  semi-diameters  of  the  two  bodies,  or  bring  the 
farthest  limbs  into  contact,  and  subtract  this  sum. 

66.  The  sextant  is  also  employed  to  take  the  altitude  of  a 
heavenly  body.  A  horizontal  reflector,  called  an  Artificial  Hori- 
zon, is  placed  in  front  of  the  observer :  the  angle  between  the 
body  and  its  reflected  image  is  then  measured,  as  if  this  image 
were  a  real  object ;  the  half  of  which  will  be  the  altitude  of  the 
body. 

A  shallow  vessel  of  mercury  forms  a  very  good  artificial 
horizon. 

67.  In  obtaining  the  altitude  of  a  body,  at  sea,  its  altitude 
above  the  visible  horizon  is  measured,  by  bringing  the  lower 
limb  into  contact  with  the  horizon.  To  this  angle  is  added  the 
apparent  semi-diameter  of  the  body,  and  from  the  result  is  sub- 

5 


34  ASTRONOMY. 

tracted  the  depression  of  the  visible  horizon  below  the  horizon- 
tal line,  called  the  Dip  of  the  Horizon. 

Micrometer. — Errors  of  Instrumental  Admeasurement. 

68.  The  Apparent  Diam,eter  of  a  heavenly  body  may  be  mea- 
sured with  great  precision  by  means  of  a  piece  of  apparatus  at- 
tached to  telescopes,  called  a  Mici^ometer,  which  is  designed  for 
the  admeasurement  of  small  angles, 

69.  AVhatever  precautions  may  be  taken,  the  results  of  instru- 
mental admeasurement  will  never  be  wholly  free  from  errors. 
Errors  that  arise  from  inaccuracy  in  the  workmanship  or  adjust- 
ment of  the  instrument  may  be  detected  and  allowed  for.  But, 
errors  of  observation  are  obviously  undiscoverable.  Since  how- 
ever the  chances  are,  that  an  error  committed  at  one  observation 
will  be  compensated  by  an  opposite  error  at  another,  it  is  to  be 
expected  that  a  more  accurate  result  will  be  obtained,  if  a  great 
number  of  observations,  under  varied  circumstances,  be  made, 
instead  of  one,  and  the  m^ean  of  the  whole  taken  for  the  element 
sought.  And  accordingly,  it  is  the  uniform  practice  of  astro- 
nomical observers  to  multiply  observations  as  much  as  is  prac- 
ticable. 


CHAPTER    IV. 

THEORY    OP    CORRECTIONS. — REFRACTION. — PARALLAX. 

ABERRATION. — PRECESSION. — NUTATION. 

70.  Angles  measured  at  the  earth's  surface  with  astronomical 
instruments,  answer  to  the  Apjyarent  Place  of  a  heavenly  body, 
and  are  termed  Apparent  elements.  In  astronomical  language, 
the  True  Place  of  a  heavenly  body  is  its  real  place  in  the  hea- 
vens, as  it  would  be  seen  from  the  centre  of  the  earth.  Angles 
which  relat  i  to  the  true  place,  are  denominated  True  elements. 
The  apparent  co-ordinates  of  a  star  are  reduced  to  the  true,  by 
the  application  of  certain  corrections,  called  Refraction,  Paral- 
lax, and  Aberration. 


REFRACTION.  35 

71.  Refraction,  and  aberration,  are  corrections  for  errors  com- 
mitted in  the  estimation  of  a  star's  place,  while  parallax  serves 
to  transfer  the  co-ordinates  from  the  earth's  surface  to  its  centre. 
The  object  of  the  reduction  of  observations  from  the  surface  to 
the  centre  of  the  earth,  is  to  render  observations  made  at  dif- 
ferent places  on  the  earth's  surface  directly  comparable  with 
each  other.  Observers  occupying  different  stations  upon  the 
earth,  refer  the  same  body  (unless  it  be  a  fixed  star)  to  different 
points  of  the  celestial  sphere.  Their  observations  cannot,  there- 
fore, be  compared  together,  unless  they  be  reduced  to  the  same 
point,  and  the  centre  of  the  earth  is  the  most  convenient  point 
of  reference  that  can  be  chosen. 

72.  The  co-ordinate  planes  or  circles,  to  which  the  place  of  a 
star  is  referred  (p.  13),  are  not  strictly  stationary,  but,  on  the 
contrary,  have  a  continual  slow  motion  with  respect  to  the 
stars.  Hence,  the  true  co-ordinates  of  a  star's  place,  which 
have  been  found  for  any  one  epoch,  will  not  answer,  without 
correction,  for  any  other  epoch.  The  reduction  from  one  epoch 
to  another,  is  effected  by  applying  two  corrections,  called  Pre- 
cession and  Nutation. 

Refraction. 

73.  We  learn  from  the  principles  of  Pneumatics,  as  well  as 
by  experiments  with  the  barometer,  that  the  atmosphere  gradu- 
ally decreases  in  density  from  the  earth's  surface  upwards.  "We 
learn  also  from  the  same  sources,  that  it  may  be  conceived  to 
be  made  up  of  an  infinite  number  of  strata,  of  decreasing 
density,  concentric  with  the  earth's  surface.  From  the  known 
pressure  and  density  of  the  atmosphere  at  the  surface  of  the 
earth,  it  is  computed,  that  by  the  laws  of  the  equilibrium  of 
fluids,  if  its  density  were  throughout  the  same  as  immediately 
in  contact  with  the  earth,  its  altitude  would  be  about  5  miles. 
Certain  facts,  hereafter  to  be  mentioned,  show  that  its  actual 
altitude  is  not  far  from  50  miles.  Now,  it  is  an  established 
principle  of  Optics,  that  light  in  passing  from  a  vacuum  into  a 
transparent  medium,  or  from  a  rarer  into  a  denser  medium,  is 
bent,  or  refracted,  towards  the  perpendicular  to  the  surface  at 
the  point  of  incidence.  It  follows,  therefore,  that  the  light 
which  comes  from  a  star,  in  passing  into  the  earth's  atmosphere, 
or  in  passing  from  one  stratum  of  atmosphere  into  another,  is 


36 


ASTRONOMY. 


refracted  towards  the  radius   drawn  from  the   centre  of  the 
eartli  to  the  point  of  incidence. 

74.  Let  M  m  m  N,  N  n  o  O,  O  o  5^  Q,  (Fis;.  15)  represent  suc- 
cessive strata  of  the  atmosphere.  Any  ray  S  p  will  then,  instead 
of  pursuing  a  straight  course  S  p  x,  follow  the  broken  line 
p  a  b  c,  being  bent  downwards  at  the  points  p,  a,  6,  c,  (fcc, 
where  it  enters  the  different  strata.  But,  since  the  number  of 
strata  is  infinite,  and  the  density  increases  by  infinitely  small 
degrees,  the  deflections  ap  x,b  a  y.,  (fcc,  as  well  as  the  lengths 
of  the  lines  p  a,  a  b,  &c.,  are  infinitely  small ;  and  therefore, 
p  a  b  c  the  path  of  the  ray,  is  a  broken  line  of  an  infinite  num- 
ber of  parts,  or  a  curved  line  concave  towards  the  earth's  sur- 
face, as  it  is  represented  in  Fig.  16.  Moreover,  it  lies  in  the 
vertical  plane  containing  the  original  direction  of  the  ray  ;  for, 
this  plane  is  perpendicular  to  all  the  strata  of  the  atmosphere, 
and  therefore  the  ray  will  continue  in  it  in  passing  from  one 
to  the  other. 

75.  The  line  O  S'  (Fig.  16),  dra^vn  tangent  to  p  a  b  the  cur- 
vilinear path  of  the  light,  at  its  lowest  point,  will  represent  the 
direction  in  which  the  light  enters  the  eye,  and  therefore  the 
apparent  line  of  direction  of  the  star.  If,  then,  O  S  be  the  true 
direction  of  the  star,  the  angle  SOS'  will  be  the  displacement  of 
the  star  produced  by  Atmospherical  Refraction.  This  angle  is 
called  the  Astronomical  Refraction,  or  simply  the  Refraction. 

Since  p  a  6  is  concave  towards  the  eartli,  O  S'  will  lie  above 
O  S  ;  consequently,  refraction  makes  the  apparent  altiiude  of  a 
star  greater  than  its  time  altitude,  and  the  apparent  zenith 
distance  of  a  star  less  than  its  true  zenith  distance.  (We  here 
speak  of  the  true  altitude  and  true  zenith  distance,  as  estimated 
from  the  station  of  the  observer  upon  the  earth's  surface.) 
Thus,  to  obtain  the  true  altitude  from  the  apparent,  we  must 
subtract  the  refraction ;  and  to  obtain  the  true  zenith  distance 
from  the  apparent,  we  must  add  the  refraction.  As  refraction 
takes  effect  wholly  in  a  vertical  plane  (Art.  74),  it  does  not 
alter  the  azim,uth  of  a  star. 

76.  The  amount  of  the  refraction  varies  with  the  apparent 
zenith  distance.  In  the  zenith  it  is  zero,  since  the  light  passes 
perpendicularly  through  all  the  strata  of  the  atmosphere ;  and 
it  is  the  greater,  the  greater  is  the  zenith  distance ;  for,  the 


REFRACTION.  37 

greater  the  angle  Z  O  a  (Fig.  17),  the  greater  will  be  the  angle  of 
refraction  O  a  C,  and  consequently  the  greater  the  refraction. 

To  find  the  amount  of  the  refraction  for  a  given  zenith  dis- 
tance or  altitude. 

77.  Let  us  first  show  a  method  of  resolving  this  problem  by 
the  general  theory  of  refraction.  According  to  this  theory,  the 
amount  of  the  refraction,  except  so  far  as  the  convexity  of  the 
strata  of  the  atmosphere  may  have  an  effect,  depends  wholly 
upon  the  absolute  density  of  the  air  immediately  in  contact  with 
the  earth,  and  not  at  all  upon  the  law  of  variation  of  the  density 
of  the  different  strata ;  that  is,  the  actual  refraction  is  the  same 
that  would  take  place,  if  the  light  passed  from  a  vacuum  imme- 
diately into  a  stratum  of  air  of  the  density  which  obtains  at  the 
earth's  surface.  Let  us  suppose  then,  that  the  whole  atmosphere 
is  brought  to  the  same  density  as  that  portion  of  it  which  is  in 
contact  with  the  earth,  and  let  6  a  ^  (F"!?-  IT")  represent  its  sur- 
face, also  let  O  represent  the  station  of  the  observer  upon  the 
earth's  surface,  and  S  a  a  ray  incident  upon  the  atmosphere  at  a. 
Denote  the  angle  of  refraction  O  a  C  by  p,  and  the  refraction  O  a 
X  by  r.     The  angle  of  incidence, 

Z'  a  S  =  Z'  a  S'  +  S'  a  S  =  O  a  C  +  O  a  .r  =p  +  r. 
Now  if  we  represent  the  index  of  refraction  of  the  atmosphere 
by  m,  we  have  by  the  laws  of  refraction, 

sin  Z'  a  S  =  m  sin  O  a  C,  or  sin  (p  -f  r)  =  tn  sinp  ; 
developing  (App.  For.  15,) 

sin  'p  cos  r  +  cos  "p  sin  r  =  m  sin  p  ; 
or  dividing  by  sin  jo, 

cos  r  -\-  cot  p  sin  r  —  m. 
But  as  r  is  small,  we  may  take  cos  r  =  1,  and  sin  r  =  r  sin  1". 
(App.  47.) 

Whence,  1  +  cot  p  r  sin  1"  =  m,  or  r  —  ^         X =  A  tang 

sm  1"       cotp 

v  ;  putting  A  =  ^^~^.      Let  Z  C  a  =  C  ;  and  Z  O  a  =  Z.    O 
^  '  ^         ''  sml"  ' 

a  C  =  Z  O  a  —  Z  C  a,  or,  p  =  Z  —  C.     Substituting,  we  have, 

r  =  A  tang  (Z  —  C)  .  .  .  .  (2). 
When  the  zenith  distance  is  not  great,  C  is  very  small  with 
respect  to  Z.     If  we  neglect  it,  we  have, 

r  =  A  tang  Z (3) ; 


S140i)7 


38  ASTRONOMY. 

which  is  the  expression  for  the  refraction,  answering  to  the 
supposition  that  the  surface  of  the  earth  is  a  plane,  and  that  the 
light  is  transmitted  through  a  stratum  of  uniformly  dense  air, 
parallel  to  its  surface.  We  perceive,  therefore,  that  the  refrac- 
tion, except  in  the  vicinity  of  the  horizon,  varies  nearly  as  the 
tangent  of  the  apparent  zenith  distance. 

78.  It  has  been  ascertained  by  experiment,  that  m  the  index 
of  refraction,  (the  barometer  being  =  29.6  inches,  and  the  ther- 
mometer =  50°)  =  1.0002803.  Substituting  in  equation  (3),  after 
having  restored  the  value  of  A,  and  reducing,  there  results, 

r  =  57".8  tang  Z (4). 

79.  Dr.  Bradley  determined  the  constant  A  of  formula  (3)  by 
means  of  astronomical  observations,  and  made  it  57".0,  which 
gives  the  formula, 

r  =  57".0  tang  Z (5). 

80.  Farther  investigations  conducted  Dr.  Bradley  to  a  more 
accurate  formula  for  the  refraction,  which  is, 

r  =  57".0  tang  (Z  —  3/-)  ....  (6)  : 
as  in  the  preceding  formula,  57"  is  here  the  refraction  at  45° 
apparent  zenith  distance.  The  value  of  r  is  obtained  by  suc- 
cessive approximations.  In  the  first  place,  r  is  considered  as 
zero,  or  37-  is  neglected  in  the  second  member ;  the  resulting  value 
found  for  the  refraction,  is  then  substituted  in  the  place  of  r  in 
the  second  member,  and  a  second  more  exact  value  computed. 

81.  Other  astronomers,  by  a  combination  of  observation  with 
theory,  have  obtained  still  more  accurate  formulae.  But  the  in- 
vestigation, as  well  as  the  explanation  of  these,  must  be  omitted 
in  an  elementary  treatise  like  the  present. 

82.  When  the  latitude  or  co-latitude  of  a  place,  and  the  polar 
distance  of  a  star  which  passes  the  meridian  near  the  zenith, 
have  been  determined,  the  refraction  may  be  found  for  all  alti- 
tudes from  observation  simply,  without  the  aid  of  theory.  For, 
let  P  (Fig.  18)  be  the  elevated  pole,  Z  the  zenith,  P  Z  E  the  me- 
ridian, H  O  R  the  horizon,  S  the  true  plaoe  of  a  star,  and  S'  its 
apparent  place.  Suppose  the  apparent  zenith  distance  Z  S'  to 
have  been  measured.  Now,  in  the  triangle  Z  P  S,  Z  P  the  co- 
latitude,  and  P  S  the  polar  distance,  are  knowii  by  hypothesis, 
and  the  angle  P  is  the  sidereal  time  which  has  elapsed  since  the 
star's  last  meridian  transit,  (or,  if  the  star  be  to  the  east  of  the 


REFRACTION.  39 

meridian,  the  difference  between  this  interval  and  24  sidereal 
hours,)  converted  into  degrees  by  allowing  15°  to  the  hour. 
Therefore  we  may  compute  the  true  zenith  distance  Z  S,  and 
subtractino-  from  it  the  apparent  zenith  distance  Z  S',  we  shall 
have  the  refraction.  For  the  solution  of  this  problem,  the  polar 
distance  may  be  found  by  taking  the  complement  of  the  declina- 
tion computed  from  an  observed  meridian  zenith  distance  (Art. 
59) ;  and,  since  the  upper  and  lower  transits  of  a  circumpolar 
star  take  place  at  equal  distances  from  the  pole,  the  co-latitude 
may  be  found  by  taking  the  half  sum  of  the  greatest  and  least 
zenith  distances  of  the  pole  star.  But  it  is  obvious,  that  neither 
of  these  quantities  can  be  accurately  determined,  unless  the 
measured  zenith  distances  be  corrected  for  refraction.  When, 
however,  the  zenith  distances  in  question  differ  considerably 
from  90°,  the  corresponding  refractions  may  be  at  first  ascer- 
tained with  considerable  accuracy  by  means  of  equation  (4). 
When  more  correct  formulae  have  been  obtained  by  this  or  any 
other  process,  the  latitude  and  polar  distance,  and  therefore  the 
refraction  answerino-  to  the  measured  zenith  distance,  will  be- 
come  more  accurately  known. 

83.  The  various  formulas  of  refraction  having  been  tested  by 
numerous  observations,  it  is  found  that  they  are  all  (though  in 
different  degrees)  liable  to  material  errors,  when  the  zenith  dis- 
tance exceeds  80°,  or  thereabouts.  At  greater  zenith  distances 
than  this,  the  refraction  is  irregular^  or  is  frequently  different  in 
amount  when  the  circumstances  upon  which  it  is  supposed  to 
depend  are  the  same. 

84.  The  refractive  power  of  the  air  varies  with  its  density, 
and  hence  the  refraction  must  vary  with  the  height  of  the  ba- 
rometer and  thermometer. 

85.  The  refractions  which  have  place  when  the  barometer 
stands  at  29.6  inches  (or,  according  to  some  astronomers,  30 
inches),  and  the  thermometer  at  50°,  are  called  mean  re- 
fractions. 

The  refractions  corresponding  to  any  other  height  of  the  ba- 
rometer or  thermometer,  are  obtained  by  seeking  the  requisite 
corrections  to  be  applied  to  the  mean  refractions,  on  the  hy- 
pothesis that  the  refraction  is  directly  proportional  to  the  density 
of  the  atmosphere. 


40  ASTRONOMY. 

86.  To  save  astronomical  obsei-vers  and  computers  the  trouble 
of  calculating  the  refraction  whenever  it  is  needed,  the  mean  re- 
fractions corresponding  to  various  zenith  distances  or  altitudes 
are  computed  from  the  formulae,  as  also  the  corrections  for  the 
barometer  and  thermometer,  and  inserted  in  a  table.  Table 
VIII  is  Dr.  Young's  table  of  mean  refractions,  and  Table  IX  his 
table  of  corrections.  The  refraction  answering  to  any  zenith 
distance  not  inserted  in  the  table  can  be  found  by  simple  propor- 
tion.    (See  Prob.  VII)*. 

Other  effects  of  atmospherical  refraction. 

87.  Atmospherical  refraction  makes  the  apparent  distance  of 
any  two  heavenly  bodies  less  than  the  true  ;  for,  it  elevates  them 
in  vertical  circles  which  continually  approach  each  other  from 
the  horizon  till  they  meet  in  the  zenith. 

88.  Refraction  also  makes  the  discs  of  the  sun  and  moon  ap- 
pear of  an  elliptical  form  when  near  the  horizon.  As  it  increases 
with  an  increase  of  zenith  distance,  the  lower  limb  of  the  sun  or 
moon  is  more  refracted  than  the  upper,  and  thus  the  vertical 
diameter  is  shortened,  while  the  horizontal  diameter  remains  the 
same,  or  very  nearly  so.  This  effect  is  most  observable  near 
the  horizon,  for  the  reason  that  the  increase  of  the  refraction  is 
there  the  most  rapid.  The  difference  between  the  vertical  and 
horizontal  diameters  may  amount  to  1-8  part  of  the  whole  dia- 
meter, 

89.  In  the  apparent  horizon  the  refraction  is  about  34'.  It 
follows,  therefore,  that  when  a  star  appears  to  be  in  the  horizon, 
it  is  actually  34'  below  it.  Refraction,  then,  retards  the  setting 
and  accelerates  the  rising  of  the  heavenly  bodies. 

Having  this  effect  upon  the  rising  and  setting  of  the  sun,  it 
must  increase  the  length  of  the  day. 

90.  The  apparent  diameter  of  the  sun  is  about  32' ;  as  this 
is  little  less  than  the  refraction  in  the  horizon,  it  follows, 
that  when  the  sun  appears  to  touch  the  horizon,  it  is  actually 
entirely  below  it.  The  same  is  true  of  the  moon,  as  its  apparent 
diameter  is  nearly  the  same  with  that  of  the  sun. 


•  The  tables  referred  to  in  the  text  may  be  found  near  the  end  of  the  book. 
The  problems  referred  to  are  in  Part  IV. 


PARALLAX.  41 

Parallax. 

91.  The  correction  for  atmospherical  refraction  having  been 
apphed,  the  zenith  distance  of  a  body  is  reduced  from  the  sur- 
face of  the  earth  to  its  centre,  by  means  of  a  correction  called 
Parallax. 

92.  Parallax  is,  m  its  most  general  sense,  the  angle  made  by 
the  lines  of  direction,  or  the  arc  of  the  celestial  sphere  comprised 
between  the  places  of  an  object,  as  viewed  from  two  different 
stations.  It  may  also  be  defined  to  be  the  angle  subtended  at 
an  object  by  a  line  joining  two  different  places  of  observation. 
Let  S  (Fig.  19)  represent  a  celestial  object,  and  A,  B  two  places 
from  which  it  is  viewed.  At  A  it  will  be  referred  to  the  point  5 
of  the  celestial  sphere,  and  at  B  to  the  point  5' ;  the  angle  B  S  A, 
or  the  arc  s  s',  is  the  parallax.  The  arc  s  s'  is  taken  as  the 
measure  of  the  angle  B  S  A,  on  the  principle  that  the  celestial 
sphere  is  a  sphere  of  an  indefinitely  great  radius,  so  that  the 
point  S  is  not  sensibly  removed  from  its  centre. 

93.  The  term  parallax  is,  however,  generally  used  in  As- 
tronomy in  a  limited  sense  only,  namely,  to  denote  the  angle 
included  between  the  lines  of  direction  of  a  heavenly  body,  as 
seen  from  a  point  on  the  earth's  surface  and  from  its  centre  ;  or 
the  angle  subtended  at  a  heavenly  body  by  a  radius  of  the  earth. 
If  C  (Fig.  20)  is  the  centre  of  the  earth,  O  a  point  on  its  surface, 
and  S  a  heavenly  body,  0  S  C  is  the  parallax  of  the  body. 

94.  The  parallax  of  a  heavenly  body  above  the  horizon,  is 
called  Parallax  in  Altitude. 

95.  The  parallax  of  a  body  at  the  time  its  apparent  altitude 
is  zero,  or  when  it  is  in  the  plane  of  the  horizon,  is  called  the 
Horizontal  Parallax.  Thus,  if  the  body  S  (Fig.  20)  be  sup- 
posed to  cross  the  plane  of  the  horizon  at  S',  O  S'  C  will  be  its 
horizontal  parallax.     O  S  C  is  a  parallax  in  altitude  of  this  body. 

96.  It  is  to  be  observed,  that  the  definition  just  given  of  the 
horizontal  parallax,  answers  to  the  supposition  that  the  earth  is 
of  a  spherical  form.  In  point  of  fact,  the  earth  (as  will  be  shown 
in  the  sequel)  is  a  spheroid,  and  accordingly  the  vertical  and  the 
radius  at  any  point  of  its  surface  are  inclined  to  each  other,  as 
represented  in  Fig.  21,  where  O  C  is  the  radius,  and  O  C  the 
vertical.  The  points  Z  and  z  in  which  the  vertical  and  radius 
pierce  the  celestial  sphere  are  called,  respectively,  the  Apparent 

6 


42  ASTRONOMY. 

Zenith,  and  the  True  Zenith.  In  perfect  strictness,  the  horizon- 
tal parallax  is  the  parallax  at  the  time  z  O  S',  the  apparent  distance 
from  the  true  zenith,  is  90°.  No  material  error,  however,  will 
be  committed,  in  supposing  the  earth  to  be  spherical,  except 
when  the  question  relates  to  the  parallax  of  the  moo7i. 

97.  Let  the  apparent  zenith  distance  Z  O  S  =  Z  (Fig.  20) 
the  true  zenith  distance  Z  C  S  =  z,  and  the  parallax  O  S  C  =  p. 
Since  the  angle  Z  O  S  is  the  exterior  angle  of  the  triangle  O  S  C, 
we  have, 

ZOS=ZCS+OSC,  and  hence  also,  Z  CS  =  ZOS— OS  C; 

or, 

Z  =  2;  +p,  and  2;  =  Z  —  p  .  .  .  .  (7). 
Thus,  to  obtain  the  true  zenith  distance  from  the  apparent,  we 
have  to  subtract  the  parallax,  and  to  obtain  the  apparent  zenith 
distance  from  the  true,  to  add  the  parallax. 

Parallax,  then,  takes  effect  wholly  in  a  vertical  plane,  like  the 
refraction,  but  in  the  inverse  manner;  depressing  the  star, 
while  the  refraction  elevates  it.  Thus,  the  refraction  is  added 
to  Z,  but  the  parallax  is  subtracted  from  it. 

To  find  an  expression  for  the  parallax  in  altitude. 

98.  1.  In  terms  of  the  apparent  zenith  distance.  In  the  tri- 
ano-le  S  O  C  (Fig.  20),  the  angle  O  S  C  =  parallax  in  altitude  = 
7>,  O  C  =  radius  of  the  earth  =  R,  C  S  =  distance  of  the  body  S  = 
D,  and  C  O  S  =  180°  —  Z  O  S  =  180°  —  apparent  zenith  dis- 
tance  =   180°  —   Z ;    and  we   have  by   Trigonometry,   the 

proportion, 

sinOSC:sinCOS::CO:CS; 

whence, 

and 

or, 


sinj:>:sin(180°— Z)  :  :R:D; 
D  sin  p  =  R  sin  Z ; 


sin  p  =  —  sin  Z  .  .  .  .  (8). 


This  equation  shows  that  the  parallax  p  depends  for  any  given 
zenith  distance  Z  upon  the  distance  of  the  body,  and  is  less  in 
proportion  as  this  distance  is  greater  :  also,  that  for  any  given 
distance  of  the  body  it  increases  with  an  increase  in  the  zenith 


PARALLAX    IN    ALTITUDE,  43 

distance.     When  Z  =  90°,  p  has  its  maximum  value,  and  then  = 

horizontal  parallax  =  H ; 

thus, 

sinH=^ (9), 

substituting, 

sin  p  =  sin  H  sin  Z  .  .  (10). 
This  last  equation  may  be  somewhat  simplified.  The  distances 
of  the  heavenly  bodies  are  so  great,  that  p  and  H  are  always 
very  small  angles  ;  even  for  the  moon,  which  is  much  the  near- 
est, the  value  of  H  does  not  at  any  time  exceed  62'.  We 
may  therefore,  without  material  error,  replace  sin  p  and  sin  H 
by  p  and  H.     This  being  done,  there  results, 

p  =  HsinZ...(ll). 
Wherefore,  the  parallax  in  altitude  equals  the  product  of  the 
horizontal  parallax  hy  the  sine  of  the  apparent  zenith  distance. 

If  we  take  notice  of  the  deviation  of  the  earth's  form  from 
that  of  a  sphere,  Z,  in  equation  (10),  will  represent  the  apparent 
distance  from  the  true  zenith,  (Art.  96,)  and  H  the  horizontal 
parallax  as  it  is  defined  in  Art.  96. 

99.  2.  In  terms  of  the  true  zenith  distance.  In  the  actual 
state  of  Astronomy,  the  true  co-ordinates  of  the  places  of  the 
heavenly  bodies  are  generally  known,  or  may  be  obtained 
by  computation,  from  the  results  of  observations  already 
made,  and  from  these  there  is  often  occasion  to  deduce  the 
apparent  co-ordinates.  For  this  purpose  there  is  required  an 
expression  for  the  parallax  in  altitude  in  terms  of  the  true 
zenith  distance. 

If  we  make  7t  =z-\-p  (Art.  97)  in  equation  (10),  we  shall 
have, 

sin  7?  =  sin  H  sin  [z  -f  »),  or  sin  H  =  __ ^HL2 —  • 
^  ^        ^'  sin  [z+p) 

whence, 

,    ,    .    TT     1    I         sin «  sin  (2;  +  79)  +  sin  p 

1  +sinH  =  l  4-  -^—r—, — -x= /     Ax? 

sm.{z-{-p)  sin  (z+p) 

and 

1  -  sinH  =  1 ^-^=  sin_(^pV-.sinp 

sin  {z-\-p)  sm  {z  -\- p) 


44  ASTRONOMY. 

dividing, 

1  -f  sin  H  _  sin  {z  -\-  p)  -{■  sin  p 
1  —  sin  iT      sin  {z  +  />)  —  sin  p  ' 

or, 

tang2  (45°  +  ^  H)  =  ^^"g^(^^+?^)  (see  App.  For.  36, 29) ; 
tang  -^  z 

whence, 

tang  ^{z  +p)  =  tang  ^  z  tang^  (45°  +  ^  H)  .  .  (12). 

This  equation  makes  known  z  -\-  p,  from  which  we  may 
obtain  p  by  subtracting  z. 

In  order  to  be  able  to  compute  the  parallax  in  altitude  by 
means  of  formula  (11)  or  (12),  it  is  necessary  to  know  H  the 
horizontal  parallax. 

To  find  the  horizontal  parallax. 

100.  Let  O,  O'  (Fig.  21)  represent  two  stations  upon  the  same 
terrestrial  meridian  O  E  O',  and  at  a  remote  distance  from  each 
other,  Z,  Z'  their  apparent  zeniths,  and  z,  z'  their  true  zeniths, 
Q,  C  E  the  equator,  and  S  the  body  (supposed  to  be  in  the  meri- 
dian,) the  parallax  of  which  is  to  be  found.  Let  the  angle  O  S  O' 
=  A,  ;^  O  S  =  Z,  ^'  O'  S  =  Z' ;  also  let  C  O  =  R,  C  O'  =  R', 
C  S  =  D,  the  parallax  in  altitude  O  S  C  =  p,  and  the  parallax 
in  altitude  0'  S  C  =  p'.  Now,  by  equation  (8),  replacing  the 
sine  of  the  parallax  by  the  parallax  itself,  (Art.  98,) 

p=  ^^m  7a,  and  p'  =  —  sin  Z' ; 

whence, 


.          ,     ,     R     .     ^  ,  R     .     „,     RsinZ+R'sin  Z' 
K^p  +y  =  _  sm  Z  +  _  sm  Z'  = -Z ..; 


but,  (equa.  9) 


H  =  _,orD  =  _. 


Substituting  this  value  of  D,  and  deducing  the  value  of  H, 
we  have, 

XT R  X  A [VS) 

RsinZ  +  R'sinZ'  '  '  '  ^     '' 
It  remains  now  to  find  an  expression  for  A  in  terms  of  mea- 
sureable  quantities.     Let  O  s  and  O'  s'  (Fig.  21,)  be  the  direc- 
tions at  O  and  O'  of  a  fixed  star  which  crosses  the  meridian 
nearly  at  the  same  time  with  the  body.     Owing  to  the  immense 


HORIZONTAL    PARALLAX.  45 

distance  of  the  star,  these  lines  will  be  sensibly  parallel  to  each 
other  (Art.  22).  Let  the  angle  S  O  5  the  difference  between  the 
meridian  zenith  distances  of  the  body  and  star,  as  observed  at  O, 
be  represented  by  rf,  and  let  the  same  difference  S  O'  s  for  the 
station  O',  be  represented  by  d'.      Now, 

OS  0'  =  0L0'  — S  0'5  =  S  O  s  —  SO'  s,0T  k  =  d~d'. 
If  the  body  be  seen  on  different  sides  of  the  star  by  the  two 
observers,  we  shall  have, 

A  =  <Z  +  </'. 
Substituting  in  equation  (13),  there  results, 

H-         ^{d±d') ,^^. 

R  sin  Z  +  R'  sin  Z'   '  *  *  ^     ^' 
If  we  regard  the  earth  as  a  sphere,  R  =  R',  and  dividing  by  R, 
we  have, 

H  =  -^-J^Jl ....  (15). 

sin  Z  +  sin  Z'  ^     ' 

101.  To  find  the  parallax  by  means  of  these  formulse,  each  of 
the  two  observers  must  measure  the  meridian  zenith  distance  of 
the  body,  and  also  of  a  star  which  crosses  the  meridian  nearly  at 
the  same  time  with  the  body,  and  correct  them  for  refraction. 
The  difference  of  the  two  will  be,  respectively,  the  values  of  d 
and  d' ;  and  the  corrected  zenith  distances  of  the  body  will  be 
the  values  of  Z  and  Z',  if  formula  (15)  be  used  ;  if  formula  (14) 
be  used,  the  measured  zenith  distances  of  the  body  must  still  be 
corrected  for  the  reduction  of  latitude,  (p.  15,  def  4.) 

It  is  not  necessary  that  the  two  stations  should  be  on  precisely 
the  same  meridian  ;  for,  if  the  meridian  zenith  distance  of  the 
body  be  observed  from  day  to  day,  its  daily  variation  will 
become  known  ;  then,  knowing  also  the  difference  of  longitude 
of  the  two  places,  a  simple  proportion  will  give  the  change  of 
zenith  distance  during  the  interval  of  time  employed  by  the 
body  in  moving  from  the  meridian  of  the  most  easterly  to  that 
of  the  most  westerly  station.  This  result,  applied  to  the  zenith 
distance  observed  at  one  of  the  stations,  will  reduce  it  to  what  it 
would  have  been,  if  the  observation  had  been  made  in  the  same 
latitude  on  the  meridian  of  the  other  station. 

102.  The  Horizontal  Parallax  of  a  heavenly  body  may  be 
found  by  the  foregoing  process,  to  within  1"  or  2"  of  the  truth. 
No  greater  degree  of  accuracy  is  necessary  in  the  case  of  the 


46  ASTROXOMV. 

moon.  But  there  are  certain  uses  made  of  the  horizontal  parallax 
of  a  body,  that  will  be  noticed  hereafter,  which  require  that  the 
parallax  of  the  sun,  and  of  the  planets,  should  be  known  with 
much  greater  precision.  The  more  accurate  methods  employed 
to  determine  the  parallaxes  of  these  bodies,  will  be  explained  (in 
principle  at  least)  in  subsequent  parts  of  the  work. 

103.  In  consequence  of  the  spheroidal  form  of  the  earth,  the 
horizontal  parallax  of  a  body  is  somewhat  different  at  different 
places.  Let  H  and  H'  denote  the  horizontal  parallaxes  of  the 
same  body,  and  R  and  R'  the  radii  of  the  earth  at  two  different 
places.     Then,  by  equation  (9), 


whence, 


H  =  |,  andH'  =  |; 


H  :  H' :  :  —  :  ?1 :  :  R :  R', 
D      D 


Thus  the  parallax  at  the  equator,  called  the  Equatorial  Par- 
allax^ is  the  greatest,  and  the  parallax  at  the  pole  the  least. 
The  difference  between  the  parallaxes  of  the  moon  at  the  equa- 
tor and  at  the  pole  may  amount  to  about  12".  For  the  other 
heavenly  bodies  the  difference  is  too  small  to  be  taken  into 
account. 

104.  Wlien  the  horizontal  parallax  has  been  found  for  any 
one  distance  and  time,  from  observation,  the  horizontal  parallax 
for  any  other  distance  and  time  may  be  approximately  computed, 
by  means  of  the  principle  that  the  parallax  of  a  body  is  directly 
proportional  to  its  apparent  diameter.  The  truth  of  this  principle 
appears  from  the  fact,  that  both  the  parallax  (Art.  98)  and  the  ap- 
parent diameter  are  inversely  proportional  to  the  same  quantity, 
viz  :  the  distance  of  the  body  from  the  centre  of  the  earth. 

In  the  present  condition  of  astronomical  science,  when  the 
horizontal  parallax  of  either  one  of  the  heavenly  bodies  is  re- 
quired for  any  particular  time,  it  may  be  obtained  by  computa- 
tion, or  from  tables.  It  may  also  be  taken  out  of  the  Nautical 
Almanac* 

*  The  Nautical  Almanac  is  a  collection  of  data  to  be  used  in  nautical  and  astro- 
nomical  calculations,  published  annually  in  England,  and  republished  in  New 
York.  It  may  generally  be  obtained  two  or  three  years  previous  to  the  year  for 
which  it  is  calculated. 


PARALLAX. ABERRATION.  47 

105.  The  equatorial  horizontal  parallax  of  the  moon  varies 
from  53'  48"  to  61'  24",  according  to  the  distance  of  the  moon 
from  the  earth.  The  equatorial  parallax  of  the  moon  answering 
to  the  mean  distance,  is  57'  1". 

The  horizontal  parallax  of  the  sun  varies  slightly  from  a 
change  of  distance.     At  the  mean  distance  it  is  8".6. 

The  horizontal  parallaxes  of  the  planets  are  comprised  within 
the  limits  31",  and  0".4. 

The  fixed  stars  have  no  parallax.* 
Parallax  in  right  ascension  a7id  declination,  and  in  longitude 

and  latitude. 

106.  Since  parallax  displaces  a  body  in  its  vertical  circle, 
which  is  generally  oblique  to  the  equator  and  ecliptic,  it  will 
alter  its  ricfht  ascension  and  declination,  as  well  as  its  longitude 
and  latitude.  The  difference  between  the  true  and  apparent 
right  ascension  is  called  the  parallax  in  right  ascension  ;  the 
like  differences  for  the  other  co-ordinates  are  called,  respectively, 
■parallax  in  declination,  parallax  in  longitude,  and  j)arallax  in 
latitude.  Formulae  by  means  of  which  these  corrections  may  be 
found,  when  the  right  ascension  and  declination  or  the  longi. 
tude  and  latitude  are  given,  are  investigated  in  the  Appendix. 

Aberration. 

107.  The  celebrated  English  Astronomer,  Dr.  Bradley,  com- 
menced in  the  year  1725  a  series  of  accurate  observations  upon 
the  fixed  stars,  which  were  continued  for  several  years,  with  the 
view  of  ascertaining  whether  the  apparent  places  of  the  fixed 
stars  were  subject  to  any  direct  alteration,  in  consequence  of 
the  supposed  change  of  the  earth's  position  in  space.  The 
observations  showed  that  there  had  been  in  reality,  during  the 
period  of  observation,  small  changes  in  the  apparent  places  of  each 
of  the  stars  observed,  which,  when  greatest,  amounted  to  about 
40" ;  but  they  were  not  such  as  should  have  resulted  from  the 
supposed  changes  of  the  earth's  position  in  space.  These  phe- 
nomena Dr.  Bradley  undertook  to  examine  and  reduce  to  a 
general  law.  After  repeated  trials,  he  at  last  succeeded  in  dis- 
covering their  true  explanation.     His  theory  is,  that  they  are 


*  The  practical  method  of  correcting  for  parallax  is  detailed  and  exemplified  in 
Problem  VIII. 


48  ASTRONOMY. 

different  effects  of  one  general  cause,  a  progressive  motion  of 
light  in  conjunction  with  an  orbitual  motion  of  the  earth. 

108.  Let  us  conceive  the  observer  to  be  stationed  at  the  earth's 
centre ;  and  let  A  C  B  (Fig.  22)  be  a  portion  of  the  earth's  orbit, 
so  small  that  it  may  be  considered  a  rigPit  line,  C  S  the  absolute 
direction  of  a  fixed  star  as  seen  from  the  point  C.AC  the  distance 
throusrh  which  the  earth  moves  in  some  small  portion  of  time, 
and  a  C  the  distance  through  which  a  particle  of  light  moves  in 
the  same  time.  Then,  a  particle  of  light,  which,  coming  from 
the  star  in  the  direction  S  C,  is  at  a  at  the  same  time  that  the 
earth  is  at  A,  will  arrive  at  E  at  the  same  time  that  the  earth 
does.  Suppose,  that  when  the  earth  is  at  A,  A  a  is  the  position 
of  the  axis  of  a  telescope,  and,  that  continuing  parallel  to  itself,  it 
takes  up  by  virtue  of  the  earth's  motion,  the  successive  positions 

A'  a',  A"  a" C  S' :  a  particle  of  light  which  follows 

the  line  S  C  in  space  will  descend  along  this  axis :  for  a  a'  is  to 
A  A'  and  a  «"  is  to  A  A",  as  a  C  is  to  A  C,  that  is,  as  the  velocity 
of  light  is  to  the  velocity  of  the  earth  ;  consequently,  when  the 
earth  is  at  A',  the  particle  of  light  is  on  the  axis  at  «',  and  when 
the  earth  is  at  A"  the  particle  of  light  is  on  the  axis  at  «",  and  so 
on  for  all  the  other  positions  of  the  axis,  until  the  earth  arrives 
at  C  The  apparent  direction  of  the  star  S,  as  far.  at  least,  as  it 
depends  upon  the  cause  under  consideration,  will  therefore  be 
CS'. 

The  angle  S  C  S',  which  expresses  the  change  in  the  apparent 
place  of  a  star  S,  produced  by  the  motion  of  light  combined  with 
the  motion  of  the  spectator,  is  called  the  Aberration  of  the  star  ; 
and  the  phenomenon  of  the  change  of  the  apparent  course  of  the 
hght  coming  from  a  star,  thus  produced,  is  called  Aberration  of 
Light,  or  simply  Aberration. 

109.  If  through  the  point  a  (Fig.  23)  a  line  a  s'  be  drawn 
parallel  to  A  C,  and  terminating  in  C  S',  the  figure  Aa  s'  G  will 
be  a  parallelogram,  and  therefore  a  s'  will  be  equal  to  A  C. 
Hence  it  appears,  that  if  on  C  S  the  line  of  direction  of  a  star 
S,  a  line  C  a  be  laid  off,  representing  the  velocity  of  light,  and 
through  a  a  line  a  s'  be  drawn,  having  the  same  direction  as  the 
earth's  motion,  and  equal  to  its  velocity,  the  line  joining  s'  and 
C  will  be  the  apparent  line  of  direction  of  the  star,  the  point  S' 
its  apparent  place  in  the  heavens,  and  the  angle  a  C  s'  its  aber- 


ABERRAT'ION.  49 

ration.  We  conclude,  therefore,  that  by  virtue  of  aberration,  a 
star  is  seen  in  advance  of  its  true  place  in  the  plane  passing 
throuo-h  the  line  of  direction  of  the  star  and  the  line  of  the 
earth's  motion. 

110.  The  aberration  is  the  same  when  a  star  is  viewed  with 
the  naked  eye,  as  when  it  is  seen  through  a  telescope.  For,  let 
a  C  the  velocity  of  the  light,  be  decomposed  into  two  velocities, 
of  which  one  A  C  is  equal  and  parallel  to  the  velocity  of  the 
earth ;  the  other  will  be  represented  by  s'  C.  Now,  since  the 
velocity  A  C  is  equal  and  parallel  to  the  velocity  of  the  earth,  it 
will  produce  no  change  in  the  relative  position  of  a  particle  of 
light  and  the  eye,  and  therefore  the  relative  motion  of  the  light 
and  the  eye  will  be  the  same  that  it  would  be  if  the  earth  were 
stationary,  and  the  light  had  only  the  velocity  s'  C  ;  accordingly, 
the  light  entering  the  eye  just  as  it  would  do  if  it  actually  came 
in  the  direction  s'  C,  and  the  eye  were  at  rest,  C  s'  will  be  the 
apparent  direction  of  the  star  from  which  it  proceeds. 

111.  If  we  regard  the  observer  as  situated  upon  the  earth's 
surface,  instead  of  being  at  its  centre,  the  aberration  resulting 
from  the  earth's  motion  of  revolution  will  be  still  the  same  :  for, 
all  points  of  the  earth  advance  at  the  same  rate  and  in  the  same 
direction  with  the  centre.  The  motion  of  rotation  will  produce 
an  aberration  proper  to  itself,  but  it  is  so  small  that  there  is  no 
occasion  to  take  it  into  account. 

112.  To  find  a  general  expression  for  the  aberration.  We 
have  by  Trigonometry  (Fig.  23), 

sin  A  a  C  :  sin  C  A  a :  :  C  A  :  C  a  :  :  vel.  of  earth  :  vel.  of  light ; 
whence, 

C  A 

sin  A  a  C  =  sin  C  A  a  — — ,  or  since  A  a  C  =  S  C  S', 
C  a 

,  •    /><  A      vel.  of  earth  /-,  e-. 

sm  aberr.  =  sm  C  A  a  — - — ^ ,.  ,     .   •  .  (151. 
vel.  of  light 

When  C  A  a  is  90°,  the  aberration  has  its  maximum  value, 

and  this  has  been  found  by  observation  to  be  about  20"  (20".36), 

whence,  .    on/;      vel.  of  earth  ,-.(.-. 

substituting,  and  taking  sin  B  C  o  for  sin  C  A  a,  to  which  it  is 
very  nearly  equal,  we  have, 

sin  aberr.  =  sin  B  C  a  sin  20"   .   .  .   (IT). 

7 


50  ASTRONOMY. 

We  may  conclude  from  this  equation  that  the  aberration  in- 
creases with  the  angle  B  C  a  made  by  tlie  direction  of  the  star 
with  the  direction  of  the  earth's  motion  ;  that  it  is  equal  to  zero, 
when  this  angle  is  zero,  and  has  its  maximum  value  of  20" 
(more  accurately  20".36)  when  this  angle  is  90°. 

113.  Let  us  now  inquire  into  the  entire  effect  of  aberration  in 
the  course  of  a  year.  Let  S  (Fig.  24)  be  the  sun  ;  E  the  earth  ;  E 
f  g  its  orbit ,  Z  T  V  that  orbit  extended  to  the  fixed  stars,  or  the 
ecliptic  (p.  11,  def  17) ;  E  T  a  tangent  to  the  earth's  orbit  at  E  ; 
O  the  place  of  S  among  the  fixed  stars  or  in  the  ecliptic,  as  seen 
from  the  earth  ;  5  a  fixed  star ;  5  V  T  the  arc  of  a  great  circle 
passing  through  5  and  T.  Then,  by  what  has  preceded  (Art. 
109),  the  earth  moving  in  the  direction  E/^,  the  apparent  place 
of  the  star  may  be  represented  by  5'  and  the  aberration  by  5  E  5'. 
Tluis,  the  effect  of  aberration  at  any  one  time  is  to  displace  the 
star  by  a  small  amount,  directly  towards  the  point  T  of  the 
ecliptic,  which  is  90°  behind  the  sun.  As  the  earth  moves,  the 
position  of  the  point  T  will  vary  ;  and  in  the  course  of  a  year, 
while  the  earth  describes  its  entire  orbit  in  the  direction  E/^, 
this  point  will  move  in  the  same  direction  entirely  around  the 
ecliptic.  In  this  period  of  time,  therefore,  5  5'  the  small  arc  of 
aberration  will  revolve  entirely  around  s  the  true  position  of  thf 
star  ;  from  which  we  conclude,  that  in  consequence  of  aberra- 
tion a  star  appears  to  describe  a  closed  curve  in  the  heavens 
around  its  true  place. 

As  the  inclination  of  the  direction  of  the  star  to  the  direction 
of  the  earth's  motion  will  vaiy  during  a  revolution  of  the  earth, 
the  aberration  will  also  vary  during  this  period  (Art.  1 12),  and 
hence  the  curve  in  question  will  not  be  a  circle.  It  appears 
upon  investigation  that  it  is  an  ellipse,  having  the  true  place  of 
the  star  for  its  centre,  and  of  which  the  semi-major  axis  is  con- 
stant and  equal  to  20".36,  and  the  semi-minor  axis  variable  and 
expressed  by  20".36  sin  X  (X  denoting  the  latitude  of  the  star). 
Each  star,  then,  describes  an  ellipse  which  is  the  more  eccentric 
in  proportion  as  the  star  is  the  nearer  to  the  ecliptic  ;  for,  the  ex- 
pression for  the  minor  axis  shows,  that  the  smaller  the  lati- 
tude the  less  will  be  this  axis.  For  a  star  situated  in  the  eclip- 
tic, the  minor  axis  will  be  zero,  and  the  ellipse  will  be  reduced 
to  a  right  line.    For  a  star  in  the  pole  of  the  ecliptic,   the 


ABERRATION.  51 

minor  axis  is  equal  to  the  major,  and  the  ellipse  therefore  be- 
comes a  circle. 

114.  Since  aberration  causes  the  apparent  place  of  a  star  to 
differ  slightly  from  its  true  place,  the  true  and  apparent  co-ordi- 
nates will,  in  consequence,  differ  somewhat  from  each  other. 
The  effects  of  the  aberration  of  light  upon  the  apparent  right 
ascension  and  declination  of  a  star,  are  called,  respectively,  the 
Aberration  in  Right  Ascension,  and  the  Aberration  in  Declina- 
tion. In  like  manner  its  effects  upon  the  longitude  and  latitude 
are  called  the  Aberration  in  Longitude,  and  the  Aberratioti  in 
Latitude.  Formulae,  for  computing  these  corrections  to  be  ap- 
plied to  the  apparent  co-ordinates  to  obtain  the  true,  or  the  re- 
verse, are  investigated  in  the  Appendix.* 

115.  Since  the  motion  of  the  earth  is  at  all  times  in  a  direc- 
tion perpendicular,  or  nearly  so,  to  the  line  followed  by  the  light 
which  comes  from  the  sun  to  the  earth,  the  aberration  of  the 
sun,  which  takes  place  only  in  longitude,  is  continually  equal 
to  20".36,  (Art.  112.)  Thus,  the  sun's  apparent  place  is  al- 
ways about  20".36  behind  its  true  place. 

116.  For  a  planet,  the  aberration  is  different  from  what  it  is 
for  a  fixed  star.  As  a  planet  changes  its  place  during  the  time 
that  the  light  is  passing  from  it  to  the  earth,  it  would,  if  the 
earth  were  stationary,  appear  to  be  as  far  behind  its  true  place 
as  it  has  moved  during  this  interval.  This  aberration  due  to 
the  motion  of  the  planet,  combined  with  that  due  to  the  earth's 
motion,  will  give  the  real  aberration  of  the  planet. 

117.  For  the  moon,  the  aberration  occasioned  by  its  motion 
around  the  earth  is  very  small.  The  earth's  motion  produces 
no  lunar  aberration,  for  the  reason  that  the  moon,  and  conse- 
quently the  light  emitted  from  it,  partakes  of  this  motion. 

118.  If  the  apparent  places  of  a  star,  found  at  various  times, 
be  corrected  for  aberration,  the  same  result  for  the  true  place  of 
the  star  is  obtained.  Again,  the  deductions  of  Art.  113  agree 
in  every  particular  with  the  observed  phenomena  of  the  appa- 
rent displacement  of  the  stars  first  discovered  by  Dr.  Bradley. 
These  facts  show  that  the  aberration  of  light  is  the  true  cause 


*  For  the  practical  method  of  determining  and  applying  these  corrections,  see 
Probs.  XIX,  XXI,  XXII,  XXIII. 


52  ASTRONOMY. 

of  these  phenomena,  and  consequently  at  the  same  time  estab- 
lish the  fact  of  the  earth's  orbitual  motion,  as  well  as  that  of 
the  progressive  motion  of  light. 

119.  It  may  be  worth  while  to  state,  that  the  first  discovery 
of  the  progressive  motion  of  light  preceded  the  detection  and 
explanation  by  Bradley  of  the  phenomena  of  aberration.  The 
discovery  was  made  by  Roemer,  a  Danish  astronomer,  in  the 
year  1667,  from  a  comparison  of  observations  upon  the  eclipses 
of  Jupiter's  satellites. 

120.  As  to  the  actual  velocity  of  light,  we  have  by  equation 
(16)  vel.  of  earth  :  vel.  of  light  :  :  sin  20"  :  1  :  :  1  :  10314.  As 
determined  from  observations  upon  Jupiter's  satellites,  it  is 
very  nearly  the  same.  The  time  employed  by  light  in  coming 
from  the  sun  to  the  earth  is  8  m.  13  s. 

Precession  and  Nutation. 

121.  In  the  investigations  that  follow,  we  shall  take  it  for 
granted  that  it  is  possible  to  find  the  obliquity  of  the  ecliptic 
and  the  place  of  the  equinox.  Methods  of  determining  them 
will  be  given,  when  we  come  to  treat  of  the  apparent  motion  of 
the  sun. 

122.  By  comparing  the  longitudes  and  latitudes  of  the  same 
fixed  stars,  obtained  at  different  periods  (Art.  60),  it  is  found  that 
their  latitudes  continue  very  nearly  the  same,  but  that  all  their 
longitudes  increase  at  the  mean  rate  of  about  50"  per  year.  The 
longitude  of  a  star  being  the  arc  of  the  ecliptic,  intercepted  in 
the  order  of  the  signs  between  the  vernal  equinox  and  a  circle 
of  latitude  passing  through  the  star  (p.  13,  def  30),  it  follows  from 
the  last  mentioned  circumstance,  that  the  vernal  equinox  must 
have  a  motion  along  the  ecliptic  in  a  direction  contrary  to  the 
order  of  the  signs,  amounting  to  about  50"  in  a  year.  As  it 
has  been  found  that  the  autumnal  equinox  is  always  at  the  dis- 
tance of  180°  from  the  vernal,  it  must  have  the  same  motion. 
This  retrograde  motion  of  the  equinoctial  points,  is  called  the 
Precession  of  the  Eqninoxes. 

123.  As  the  latitude  of  a  star  is  its  distance  from  the  ecliptic 
(p.  13,  def  31),  it  follows  from  the  circumstance  of  the  latitudes 
of  all  the  stars  continuing  very  nearly  the  same,  that  the  ecliptic 
remains  fixed,  or  very  nearly  so,  with  respect  to  the  situations 
of  the  fixed  stars. 


PRECESSION.  53 

124.  The  ecliptic  being  stationary,  it  is  plain  that  the  preces- 
sion of  the  equinoxes  must  result  from  a  continual  slow  motion 
of  the  equator  in  one  direction.  It  appears  from  observation, 
that  the  obliquity  of  the  ecliptic,  or  the  inclination  of  the  equa- 
tor to  the  ecliptic,  remains,  in  the  course  of  this  motion,  very 
nearly  the  same. 

125.  Since  the  equator  is  in  motion,  its  pole  must  change  its 
place  in  the  heavens.  Let  V  L  A  (Fig.  25)  represent  the  eclip- 
tic, K  its  pole,  which  is  stationary,  P  the  position  of  the  north 
pole  of  the  equator  or  of  the  heavens  at  any  given  time,  and 
V  E  A  the  corresponding  position  of  the  line  of  the  equinoxes  : 
K  P  L  represents  the  circle  of  latitude  passing  through  P,  or  the 
solstitial  colure.  Now,  the  point  V  being  at  the  same  time  in 
the  ecliptic  and  equator,  is  90°  distant  from  the  two  points 
K  and  P  the  poles  of  these  circles  ;  therefore,  it  is  the  pole  of  the 
circle  K  P  L  passing  through  these  points,  and  hence  V  L  =  90°. 
It  follows  from  this,  that  when  the  vernal  equinox  has  retro- 
graded to  any  point  V,  the  pole  of  the  equator,  originally  at  P, 
will  be  found  in  the  circle  of  latitude  K  P'  L'  for  which  V  L' 
equals  90°  :  it  will  also  be  at  the  distance  K  P'  from  the  pole  of 
the  ecliptic  equal  to  K  P.  Whence  it  appears,  that  the  pole  of 
the  equator  has  a  retrograde  motion  in  a  small  circle  about  the 
pole  of  the  ecliptic,  and  at  a  distance  from  it  equal  to  the  obli- 
quity of  the  ecliptic.  As  the  motion  of  the  equator  which  pro- 
duces the  precession  of  the  equinoxes,  is  uniform,  the  motion  of 
the  pole  must  be  uniform  also  ;  and  as  the  pole  will  accomplish 
a  revolution  in  the  same  time  with  the  equinox,  its  rate  of 
motion  must  be  the  same  as  that  of  the  equinox,  that  is, 
50"  of  its  circle  in  a  year.  The  period  of  the  revolution  of 
the  equinox  and  of  the  pole  of  the  equator  is  25920  years 
/     360°\ 

V~  "50^/ 

126.  The  ecliptic,  although  very  nearly  stationary,  as  stated 
in  Art.  123,  is  not  strictly  so.  By  comparing  the  values  of  the 
obliquity  of  the  ecliptic,  found  at  distant  periods,  it  is  ascertained 
that  it  is  subject  to  a  gradual  diminution  from  century  to  century. 
A  comparison  of  the  results  of  observations  made  by  Flamstead 
in  1690,  and  by  Dr.  Maskelyne  in  1769,  gives  for  the  mean 
secular  diminution  50",  and  for  the  mean  annual  diminution 


54  ASTRONOMY. 

0".50.     A  more  accurate  determination  of  the  mean  annual 
diminution  is  0".46. 

It  appears  from  observation,  that  there  are  minute  secular 
changes  in  the  latitudes  of  the  stars,  which  establish  that  the 
diminution  of  the  obliquity  of  the  ecliptic  arises  from  a  slow  dis- 
placement of  the  plane  of  the  ecliptic  (or  of  the  earth's  orbit)  in 
space. 

127.  If  the  ecliptic  slowly  changes  its  position  in  the 
heavens,  its  pole  must  likewise  ;  and  since  the  obliquity  of  the 
ecliptic  is  continually  diminishing,  its  pole  must  be  gradually 
approaching  the  pole  of  the  equator. 

128.  The  motion  of  the  ecliptic  alters  somewhat  the  precession 
of  the  equinoxes,  making  it  a  little  less  than  it  would  be,  if  the 
equator  only  was  in  motion  :  for,  let  E  L  (Fig.  26)  represent  the 
position  of  the  ecliptic,  and  V  d  that  of  the  equator,  at  any 
assumed  date,  and  E  L',  V  Q,'  the  positions  of  the  same  circles 
at  some  later  date ;  the  obliquity  L'  V"  Q,'  at  the  second  epoch 
being  less  than  that  (L  V  Q.)  at  the  first  epoch  :  also  let  v  be 
the  physical  point  of  the  moveable  ecliptic,  which  at  the  first 
epoch  coincided  with  the  point  V.  If  the  ecliptic  remained  sta- 
tionary in  the  position  E  L,  the  precession  during  the  interval  of 
the  epochs  would  be  V  V.  But,  by  reason  of  its  motion,  the 
actual  precession  is  v  V",  and  it  is  obvious  from  an  inspection 
of  the  figure,  (the  angle  V  V  Q.'  being  an  acute  angle,)  that  this 
is  less  than  V  V  the  precession  on  the  fixed  ecliptic.  We  learn 
by  the  aid  of  Physical  Astronomy,  that  the  amount  of  annual 
precession  would,  if  the  ecliptic  were  fixed,  be  50". 35.  As  we 
have  already  seen,  the  actual  precession  on  the  moveable  ecliptic 
is  50"  (more  accurately,  50".23). 

129.  The  motion  of  the  equator  which  produces  the  preces- 
sion of  the  equinoxes,  must  also  produce  changes  in  the  rio-ht 
ascensions  and  declinations  of  the  stars.  These  chansfes  will 
be  different,  according  to  the  situations  of  the  stars  with  respect 
to  the  equator  and  equinoctial  points. 

130.  It  remains  for  us  now  to  take  notice  of  a  minute  ine- 
quality in  the  motion  of  the  equator  and  its  pole,  which  we  have 
thus  far  overlooked.  Dr.  Bradley,  in  observing  the  polar  dis- 
tance of  a  certain  star,  with  the  view  of  verifying  his  theory  of 
aberration,  discovered  that  the  observed  polar  distance  did  not 


NUTATION.  55 

agree  with  the  apparent  polar  distance,  as  computed  from  the 
results  of  previous  observation,  by  allowing  for  precession,  aber- 
ration, and  refraction  ;  and  hence  inferred  the  existence  of  a  new 
cause  of  variation  in  the  co-ordinates  of  a  star.  On  continuing 
his  observations,  he  found  that  the  polar  distance  alternately 
increased  and  diminished,  and  that  it  returned  to  the  same  value 
in  about  19  years.  These  phenomena  led  him  to  suppose  that 
the  pole,  instead  of  moving  uniformly  in  a  circle  around  the  pole 
of  the  ecliptic,  revolved  around  a  point  conceived  to  move  in  this 
manner. 

If  the  pole  has  such  a  motion,  it  is  plain  that  (allowing  the 
fact  of  the  earth's  rotation)  it  must  result  from  a  vibratory  mo- 
tion of  the  earth's  axis.  To  this  supposed  vibration  of  the  axis 
of  the  earth,  and  consequently  of  that  of  the  heavens.  Dr.  Brad- 
ley gave  the  name  of  Nutation.  The  term  Nutation  is  also  ap- 
plied to  the  changes  of  the  co-ordinates  of  a  star's  place,  which 
are  produced  by  the  nutation  of  the  earth's  axis.  The  point 
about  which  the  pole  was  conceived  to  revolve,  is  the  mean  po- 
sition of  the  pole,  or  the  Mean  Pole. 

Dr.  Bradley  discovered,  from  his  observations,  that  the  curve 
described  by  the  pole  must  be  an  ellipse,  having  its  major  axis 
in  the  solstitial  colure ;  and  estimated  the  value  of  the  major 
axis  at  about  18",  and  that  of  the  minor  axis  at  about  16".  He 
also  discovered  that  a  connection  existed  between  the  position  of 
the  pole  in  its  ellipse,  and  the  position  of  the  moon  at  the  time  its 
latitude  was  zero  (Art.  60),  and  changing  from  south  to  north, 
or  of  the  point  in  which  the  moon  crossed  the  plane  of  the  eclip- 
tic, in  passing  from  the  south  to  the  north  side  of  it,  called  the 
ascending  node  of  the  moon's  orbit ;  for,  he  found  that  the  pole 
retrograded  in  like  manner  with  the  node  ;  that  it  completed  its 
revolution  in  the  same  time,  namely,  in  about  19  years ;  and 
that  its  position  was  determinable  from  the  place  of  the  node 
by  a  geometrical  construction.  Let  P  (Fig.  27)  represent  the 
mean  pole,  and  j)  the  true  pole  ]  p  f  g  represents  the  ellipse 
described  by  the  true  pole  around  P  as  a  centre  ;  g  g',  lying  in 
the  solstitial  colure  K  P  L,  being  its  major  axis,  and  //'  its 
minor  axis.  It  is  to  be  observed,  that  the  pole  P  is  not  station- 
ary, but  revolves  in  the  circle  N  P  P',  carrying  with  it  the  el- 
lipse p/^. 


56  ASTRONOMY. 

Tliis  theory  of  a  nutation  of  the  earth's  axis  has  been  com- 
pletely verified  by  subsequent  observations,  and  Physical  Astro- 
nomy has  revealed  the  cause  of  the  phenomenon. 

131.  As  the  equator  must  move  with  the  axis  of  the  earth  or 
heavens,  nutation  will  change  the  position  of  the  equinox  and 
the  obliquity  of  the  ecliptic.  It  is  plain  that  its  effect  upon  the 
position  of  the  equinox  will  be  to  make  it  oscillate  periodically, 
and  by  equal  degrees,  from  one  side  to  the  other  of  the  position 
which  corresponds  to  the  mean  pole ;  and  that  its  effect  upon 
the  obliquity  of  the  ecliptic,  will  be  to  make  it  alternately 
greater  and  less  than  the  obliquity  corresponding  to  the  mean 
pole.  The  position  of  the  equinox  which  corresponds  to  the 
mean  pole,  is  called  the  Mean  Equinox.  The  obliquity  corres- 
ponding to  the  mean  pole,  is  termed  the  Mean  Obliquity. 
Mean  Equator  has  a  like  signification.  The  real  equinox  and 
the  real  equator  are  called,  respectively,  the  True  Equinox  and 
the  True  Equator.  The  actual  obliquity  of  the  ecliptic  is 
termed  the  Apparent  Obliquity.  Right  ascension  and  de- 
clination, as  estimated  from  the  true  equator  and  true  equinox, 
are  called,  respectively.  True  Right  Ascension^  and  Tnie  De- 
clination ;  and  longitude,  as  reckoned  from  the  true  equinox,  is 
called  True  Longitude.  flight  ascension,  declination,  and 
longitude,  reckoned  from  the  mean  equinox  and  mean  equator, 
are  called,  respectively,  Mean  Right  Ascension,  Mean  Declina- 
tion, and  Mean  Longitude.  The  true  and  mean  co-ordinates 
differ  by  reason  of  nutation.  The  effect  of  nutation  upon  the 
right  ascension  is  called  the  Nutation  in  Right  Ascension; 
upon  the  declination.  Nutation  in  Declination  ;  and  upon  the 
longitude.  Nutation  in  Longitude.  Its  effect  upon  the  obliquity 
of  the  ecliptic  is  called  Nutation  of  Obliquity.  The  distance  of 
the  true  from  the  mean  equinox  in  longitude,  which  is  the  same 
as  the  nutation  in  longitude,  is  sometimes  termed  the  EquatioJt 
of  the  Equinoxes  in  Longitude;  and  the  distance,  in  right 
ascension,  the  Equation  of  the  Equinoxes  in  Right  Ascension. 
The  precession  of  the  mean  equinox  is  equal  to  the  Mean  Pre- 
cession of  the  true  equinox,  which  is  50".2. 

132.  Formulae,  for  computing  the  nutation  in  right  ascension, 
declination,  &c.  at  any  given  time,  are  investigated  in  the  Ap- 
pendix.    These  formulae  cannot  be  used  without  a  knowledge  of 


REDUCTION    OF    CO-ORDINATES.  Oi 

the  moon's  motions.  In  practice,  the  nutations  in  right  ascen- 
sion, (fcc.  are  found  by  the  aid  of  tables.  (See  Probs.  XX,  XXIII.) 
If  these  be  apphed  to  the  true  co-ordinates,  the  resuUs  will  be 
the  mean  co-ordinates.  If  the  mean  co-ordinates  be  known,  the 
same  corrections  will  furnish  the  true. 

133.  Physical  Astronomy  has  made  known  the  existence  of 
another  nutation  of  the  earth's  axis,  too  small  to  be  detected  by 
observation.  It  is  called  Solar  Nutation.  The  nutation  disco- 
vered by  Dr.  Bradley  is  frequently  called  Lunar  Nutation. 

To  reduce  the  co-ordinates  of  a  star  from  one  epoch  to  another. 

134.  This  problem  is  resolved  by  first  converting  the  true  co- 
ordinates into  the  mean,  then  transferring  the  mean  co-ordinates 
from  the  one  epoch  to  the  other,  and  finally  converting  the  re- 
duced mean  co-ordinates  into  the  true.  The  mode  of  performing 
the  first  and  last  mentioned  operations  has  already  been  consi- 
dered (Art.  132).  It  remains  now  for  us  to  show  how  to  reduce 
mean  co-ordinates  froin  one  epoch  to  another. 

135.  1.  When  the  interval  of  time  between  the  epochs  compri- 
ses hut  a  few  years.  In  this  case  the  changes  from  precession,  of 
the  mean  right  ascension  and  declination  in  the  course  of  a  year, 
called  the  Annual  Variation  in  right  ascension,  and  the  Annual 
Variation  in  declination,  are  determined,  then  multiplied  by  the 
number  of  years  in  the  interval,  and  applied  as  corrections  to  the 
given  right  ascension  and  declination. 

For  this  purpose  formulae  have  been  investigated,  in  which 
the  annual  variations  in  right  ascension  and  declination  are  ex- 
pressed in  terms  of  the  right  ascension  and  declination  of  the 
star,  and  the  obliquity  of  the  ecliptic.  Let  V  L  A  (Fig.  28)  be 
the  ecliptic,  K  its  pole,  P  P'  P"  the  circle  described  by  the 
mean  pole,  P  the  mean  pole  and  V  Q,  A  the  mean  equator  at 
any  given  time,  P'  the  mean  pole  and  V  QJ  A'  the  mean 
equator  a  year  afterwards,  and  s  a  star.  Draw  P'  r  perpendicular 
to  the  declination  circle  V  s  a.     We  have, 

an.  var.  in  dec.  =  s  a'  —  s  a  =  V  s  —  P'5  =  Pr; 
but  since  P  P'  r  may  be  considered  as  a  right  angled  plane  triangle, 

P  r  =  P  P'  cos  P'  P  r  =  P  P'  sin  a  P  «  .  .  .  (18). 
Regarding  K  P  P'  as  a  right  angled  isoceles   triangle,   we 
obtain, 

sin  K  P  P'  or  1  :  sin  K  P' : :  sin  P  K  P'  :  sin  P  P' ; 
8 


58  ASTRONOMY. 

whence, 

sin  P  P'  =  sin  P  K  P'  sin  K  P,  or  P  P'  =P  K  P'  sin  K  P' 

(nearly)  .  .  .  (19) ; 
substituting  in  equation  (18),  there  results, 

P  r  =  P  K  P'  sin  K  P'  sin  a  P  a. 
P  K  P'  =  50".2  (Art.  125) ;  K  P'  =  obliquity  of  the  ecliptic  =  w  ; 
ClPa  =  VQ,  —  Va  =  90°  —  R  (R  designating  the  right  ascen- 
sion of  the  star  s).     Thus,  finally, 

an.  var.  in  dec.  =  50".2  sin  u  cos  R  ,  ,  .  (20). 
Next,  we  have, 
an.  var.  in  r.  asc.  =  V  a'  —  \  a^Y'  a'  —  mb  =  \'  m-{-b  a' .  .  (21), 
but,  V  m  =  V  V  cos  V  V  m  =  50".2  cos  w ; 

and  since  the  right  angled  triangles  5  P'  r  and  s  b  a'  are  similar, 

sin  5  r  or  sin  s  P'  (nearly)  :  sin  P'  r  :  :  5  a'  :  sin  b  a' ; 
whence, 

sin  ba'  =  sin  P'  r  !HLii^',  oYba'  =  V'r  ^^"  ^  ^'  (nearly). 
sinP's  sinP's^         ^^ 

The  triangle  P  P'  r  gives  P'  r  =  P  P'  sin  P'  P  r  =  P  P'  cos  Q  P  a 

=  P  K  P'  sin  K  P'  cos  a  P  a  (equa.  19)  ;  and  sin  P'  5  =  cos  5  a'. 

Substituting,  we  obtain 

6  a'  =P  KF  sin  K  P'  cos  a  P  a  ^"^  ^  ""'   =  PK  P'  sin  K  F  cos 

cos  s  a' 

Q,  P  a  tang  s  a'. 
Replacing  P  K  P',  K  P',  and  Gl  P  a  by  their  values,  as  above,  and 
taking  the  declination  s  a  for  s  a'  and  denoting  it  by  D,  there 
results, 

b  a'  =  50".2  sin  u  sin  R  tang  D. 
Now,  substituting  in  equation  (21)  the  values  of  V  m  and  b  a', 
we  have, 

an.  var.  in  r.  asc.  =  50".2  cos  w  -j-  50".2  sin  w  sin  R  tang  D  .  .  (22). 
The  results  of  formulas  (20, 22)  are  to  be  used  with  their  alge- 
braic signs,  if  the  reduction  is  from  an  earlier  to  a  later  epoch, 
otherwise  with  the  contrary  signs.  The  declination  is  always 
to  be  considered  jwsitive,  if  North,  and  negative,  if  South. 

V  m  =  50".2  cos  u  =  50".2  cos  23°  28'  =  46".0 
is  the  annual  retrograde  motion  of  the  equinoctial  points  along 
the  equator. 

_    136.  2.  Wlien  the  interval  of  the  epochs  is  of  considerable  or 
great  length.    If  the  epochs  are  separated  by  an  interval  of  more 


VARIATIONS    OF    THE    CORRECTIONS.  59 

than  10  or  12  years,  the  foregoing  process  will  not  answer :  for 
in  a  period  often  years  the  annual  variations  will  have  sensibly 
altered.*  In  this  case  we  may  proceed  as  follows  :  Convert  the 
right  ascension  and  declination  into  longitude  and  latitude,  add 
to  the  longitude  (or  if  the  reduction  be  to  an  earlier  epoch,  sub- 
tract from  it)  the  precession  in  longitude,  which  will  be  the 
product  of  50". 23  by  the  interval  of  the  epochs,  expressed  in 
years  and  parts  of  a  year,  and  then  with  the  longitude  thus 
obtained,  and  the  latitude,  calculate  the  right  ascension  and  decli- 
nation, using  the  mean  obliquity  of  the  ecliptic. 

When  the  period  is  of  great  length,  or  very  great  precision  is 
desired,  the  precession  on  the  fixed  ecliptic  should  be  used,  which 
is  50". 35  per  year  (Art.  128) ;  and  the  right  ascension  should  be 
corrected  for  the  change  of  the  position  of  the  equinox  on  the 
equator,  produced  by  the  motion  of  the  ecliptic  ;  which  correc- 
tion is  —  0".13  (per  year)  for  later  epochs. 
Remarks  on  the  Corrections. —  Verification  of  the  Hyjjothesis  that 

the  Diurnal  Motion  of  the  Stars  is  Uniform  and  Circular. 

137.  It  appears  from  what  we  have  stated  on  the  subject  of  the 
Corrections  :  1.  That  Refraction  varies  during  the  day  with  the 
altitude  of  the  body,  and  changes  for  all  altitudes  with  the  state 
of  the  atmosphere  ;  2.  That  Parallax  varies,  like  the  Refraction, 
with  the  altitude  of  the  body,  and  changes  from  one  day  to 
another  with  its  distance  ;  3.  That  Aberration  remains  sensibly 
the  same  for  two  or  three  days,  and  depends  for  its  absolute  value 
on  the  time  of  the  year  ;  4.  That  Precession  and  Nutation  do  not 
perceptibly  alter  the  co-ordinates  of  a  star,  unless  it  be  a  circum- 
polar  star,  under  several  days,  and  that  the  former  increases 
uniformly  with  the  time,  while  the  latter  varies  periodically,  its 
effects  entirely  disappearing  in  about  19  years  ;  and  5.  That  the 
absolute  value  of  the  Nutation  depends  entirely  upon  the  longi- 
tude of  the  moon's  ascending  node. 

138.  In  the  determination  of  the  amount  and  laws  of  the  cor- 
rections, it  was  taken  for  granted  by  astronomers,  that  the  diurnal 


*  It  is  to  be  understood  that  we  are  here  giving  methods  of  obtaining  very 
accurate  results.  The  process  just  explained,  except  for  stars  near  the  pole,  will 
furnish  results  sufficiently  accurate  for  most  purposes,  even  when  the  interval 
comprises  20  years  or  more. 


60  ASTRONOMY. 

motion  of  the  stars  was  uniform  and  circular.  This  hypothesis 
may  be  verified  in  the  following  manner :  Let  the  zenith  dis- 
tance and  azimuth  of  the  same  star  be  measured  at  various  times 
during  a  revolution,  and  corrected  for  refraction  (the  other  correc- 
tions being  insensible,  Art.  137).  Then,  if  the  latitude  of  the 
place  be  known  (Art.  59),  in  the  triangle  Z  P  S  (Fig.  13)  we 
shall  have  Z  P  the  co-latitude,  Z  S  the  zenith  distance  of  the 
star,  and  P  Z  S  its  azimuth,  whence  we  may  compute  P  S.  If 
this  calculation  be  made  for  the  time  of  each  observation,  it  will 
be  found  that  the  same  value  for  P  S  is  obtained  in  every  instance  ; 
which  proves  the  diurnal  motion  to  be  circular.  Again,  let  the 
angle  Z  P  S  be  computed  for  the  time  of  each  observation,  with 
the  same  data,  and  it  will  be  found  that  it  varies  proportionally  to 
the  time ;  which  establishes  that  the  diurnal  motion  is  also  uni- 
form, or,  at  least,  sensibly  so  during  one  revolution. 

139.  When  the  transits  of  a  circumpolar  star  are  observed  at  in- 
tervals of  several  days,  and  allowance  is  made  for  the  error  of  the 
rate  of  the  clock,  as  determined  from  observations  upon  stars  in 
the  vicinity  of  the  equator,  and  for  the  aberration  in  right  ascen- 
sion, it  is  found  that  the  times  of  the  transits  differ  slightly  from 
each  other ;  from  which  it  appears,  that  the  diurnal  motion  of  the 
stars  is  not  strictly  uniform.  When,  however,  allowance  is  made 
for  the  precession  and  nutation  in  right  ascension,  this  difference 
disappears.  We  may  hence  conclude  that  the  motion  of  rotation 
of  the  earth  is  uniform,  and  that  the  motions  of  the  earth  and  of  its 
axis,  which  produce  the  phenomena  of  precession  and  nutation, 
alter  the  times  of  the  transits  of  the  stars,  thereby  making  the  pe- 
riod of  the  apparent  revolution  of  a  star  to  differ  slightly  from  the 
period  of  the  earth's  rotation. 

It  may  be  observed,  that  the  greatest  difference  obtains  in  the 
case  of  the  pole  star,  and  is  half  a  second. 


FIGURE    AND    DIMENSIONS    OF    THE    EARTH.  61 


CHAPTER    V. 

OF    THE    EARTH  ; ITS    FIGURE    AND    DIMENSIONS  : — LATI- 
TUDE   AND    LONGITUDE    OF    A    PLACE. 

140.  Although  it  is  in  general  sufficient  for  astronomical  pur- 
poses, to  regard  the  earth  as  a  sphere,  still  it  is  necessary  in  some 
cases  of  astronomical  observation  and  computation,  when  accu- 
rate results  are  desired,  to  take  notice  of  its  deviation  from  the 
spherical  form.  No  account  need,  however,  be  taken  of  the 
irregularities  of  its  surface,  occasioned  by  mountains  and  valleys, 
as  they  are  exceedingly  minute  when  compared  with  the  whole 
extent  of  the  earth.  It  is  to  be  understood,  then,  that  by  the  figure 
of  the  earth  is  meant  the  general  form  of  its  surface,  supposing  it 
to  be  smooth,  or  that  the  surface  of  the  land  corresponded  with 
that  of  the  sea. 

141.  The  figure  of  the  earth  is  ascertained  from  an  examina- 
tion of  the  form  of  the  terrestrial  meridians. 

A  Degree  of  a  terrestrial  meridian  is  an  arc  of  it  corresponding 
to  an  inclination  of  1°  of  the  verticals  at  the  extremities  of  the  arc. 
It  is  also  called  a  Degree  of  Latitude. 

142.  The  length  of  a  degi-ee  at  any  place  will  serve  as  a  mea- 
sure of  the  curvature  of  the  meridian  at  that  place ;  for  it  is  ob- 
vious from  considerations  already  presented  (Art.  2),  that  the 
earth,  if  not  strictly  spherical,  must  be  nearly  so,  and  therefore 
that  a  degree  a  h  (Fig.  29)  may,  with  but  little,  if  any  error,  be 
considered  as  an  arc  of  1°  of  a  circle,  which  has  its  centre  at  C 
the  point  of  intersection  of  the  verticals  C  a,  C  6  at  the  extremities 
of  the  arc.  The  curvature  will  then  decrease  in  the  same  propor- 
tion as  the  radius  of  this  circle  increases,  and  therefore  in  the 
same  proportion  as  the  length  of  a  degree  increases.  Wherefore, 
the  form  of  a  meridian  may  be  determined  by  measuring  the 
length  of  a  degree  at  various  latitudes. 

143.  To  determine  the  length  of  a  degree  of  a  terrestrial 
meridian.    To  accomplish  this,  we  have, 

1.  To  run  a  meridian  line  ;  an  operation  which  is  performed  in 


62 


ASTRONOMY. 


the  following  manner :  An  altitude  and  azimuth  instrument  (or 
some  other  •instrument  adapted  to  meridian  observations)  is  first 
placed  at  the  point  of  departure,  and  accurately  adjusted  to  the 
meridian.  A  new  station  is  then  established  by  sighting  forward 
with  the  telescope.  To  this  station  the  instrument  is  Removed, 
and  is  there  adjusted  to  the  meridian  by  sighting  back  to  the  first 
station.  A  third  station  is  then  established  by  sighting  forward 
with  the  telescope  as  before,  to  which  the  instrument  is  removed. 
By  thus  continually  establishing  new  stations,  and  carrying  the 
instrument  forward,  the  meridian  line  may  be  marked  out  for  any 
required  distaino.  The  meridian  adjustments  may  be  corrected 
from  time  to  time  by  astronomical  observations  (Arts.  45,  62). 

2.  To  find  the  length  of  the  arc  passed  over.  When  the  gi'ound 
is  level,  the  length  of  the  arc  it  ay  be  directly  measured.  In  case 
the  nature  of  the  gi'ound  is  such  as  not  to  allow  of  a  direct 
measurement,  it  may  be  calculated  with  equal  precision,  by  means 
of  a  base  line,  and  a  chain  of  triangles  the  angles  of  which  are 
measured. 

3.  To  find  the  inclination  of  the  verticals  at  the  extreme  sta- 
tions. This  angle  may  be  obtained  by  measuring  the  meridian 
zenith  distances  of  the  same  fixed  star  at  the  two  stations,  correct- 
ing them  for  refraction  if  they  are  observed  about  the  same  time  ; 
and  for  refraction,  aberration,  precession,  and  nutation,  if  they  are 
observed  at  different  times,  and  taking  their  difference.  For,  let 
O,  O'  (Fig.  29)  be  the  two  stations  in  question,  Z,  Z'  their  zeniths, 
and  O  S,  O'  S  the  directions  of  a  fixed  star,  and  we  shall  have, 

OcO'  =  ZOI  —  OIc=ZOS  —  Z'IS  =  ZOS  —  Z'O'S; 
that  is,  the  angle  comprised  between  the  verticals  equal  to  the 
difference  of  the  meridian  zenith  distances  of  the  same  star. 

4.  The  length  of  an  arc  of  the  tneridian,  either  someichat 
greater  or  less  than  a  degree^  having  been  found  hy  the  fore- 
going operations,  thence  to  compute  the  length  of  a  degree. 
Let  N  denote  the  number  of  degrees  and  parts  of  a  degree  in  the 
measured  arc,  A  its  length,  and  x  the  length  of  a  degree.  Then, 
allowing  that  the  earth  for  an  extent  of  several  degrees  does  not 
difier  sensibly  from  a  sphere,  we  may  state  the  proportion, 

N  :  A  : :  1°  :  :r,  whence  x  =  ^°  ^  ^    .  .  .  (23). 
144.  Degrees  have  been  measured  with  the  greatest  possible 


FIGURE    AND    DIMENSIONS    OF    THE    EARTH.  63 

care,  at  various  latitudes  and  on  various  meridians.  Upon  a 
comparison  of  the  measured  degi-ees,  it  a  ipc  ars  that  the  length  of 
a  degree  increases  as  loe  froceed  from  the  equator  towards  either 
jwle.  It  follows,  therefore,  (Art.  132,)  that  the  curvature  of  a 
meridian  is  greatest  at  the  equator,  and  diminishes  as  we  go 
towards  the  poles  ;  and,  consequently,  that  the  earth  is  flattened 
at  the  poles. 

145.  The  fact  of  the  decrease  of  the  curvature  of  a  terrestrial 
meridian  from  the  equator  to  the  poles,  leads  to  the  supposition 
that  it  is  an  ellipse,  having  its  major  axis  in  the  plane  of  the  equa- 
tor, and  its  minor  axis  coincident  with  the  axis  of  the  earth. 
Analytical  investigations,  founded  on  the  lengths  of  a  degree  in 
different  latitudes  and  on  different  meridians,  prove  that  a  meri- 
dian is,  in  fact,  very  nearly  an  ellipse,  and  that  the  earth  has  very 
nearly  the  form  of  an  oblate  spheroid.  The  same  investigations 
also  make  known  the  dimensions  of  the  earth.  The  amount  of 
the  oblateness  at  the  poles  is  measured  by  the  ratio  of  the  differ- 
ence of  the  equatorial  and  polar  diameters  to  the  equatorial 
diameter,  which  is  technically  termed  the  Oblateness. 

146.  The  form  of  the  earth  has  also  been  determined  by  other 
methods,  which  cannot  here  be  explained.     All  the  results,  taken 

together,  indicate  an  oblateness  of -, 

305 

The  following  are  the  dimensions  of  the  earth  in  miles  : 

Radius  at  the  equator 3962.6  miles. 

Radius  at  the  pole 3949.6       " 

Difference  of  equatorial  and  polar  radii         13.0       " 

Mean  radius,  or  at  45°  latitude,  .     .     .     3956.1       " 

Mean  length  of  a  degree 69.05     " 

The  fourth  part  of  a  meridian    .     .     .     6214.2       " 

147.  Owing  to  the  elliptical  form  of  a  terrestrial  meridian,  the 
radius  and  vertical  at  a  place  do  not  coincide.  Let  E  N  Gl  S  (Fig. 
30)  represent  a  terrestrial  meridian.  For  any  point  O  situated 
on  this  meridian,  C  O  will  be  the  radius,  and  the  normal  line 
Z  O  N  the  vertical.  The  position  of  the  verticnl  will  always  be 
such,  that  the  apparent  zenith  Z  will  lie  between  the  true 
zenith  z  and  the  elevated  pole  P.  The  inclination  of  the  radius  to 
the  vertical,  or  the  angle  CON,  called  the  reduction  of  latitude, 
is  greatest  at  the  latitude  45°,  and  is  there  equal  to  about  11^'. 


64  ASTRONOMY. 

148.  The  oblatencss  of  the  earth  occasions  some  shght  modifi- 
cations in  the  effects  of  parallax,  which  are  in  some  instances  to  be 
taken  into  account  in  computing  the  apparent  azimuth  and  zenith 
distance  of  a  body,  from  the  Imown  co-ordinates  of  its  true  place. 
(These  are  investigated  in  the  Appendix.) 

Determination  of  the  Latitude  and  Longitude  of  a  Place. 

149.  The  latitude  and  longitude  of  a  place  ascertain  its  situa- 
tion upon  the  earth's  surface,  and  are  essential  elements  in  many 
astronomical  investigations. 

150.  To  find  the  latitude  of  a  place. 

1.  Bi/  the  zenith  distances  or  altitudes  of  a  circumpolar  star 
at  its  upper  and  lovjer  transits.  The  principle  of  this  method 
has  already  been  demonstrated  (Art.  59),  and  shown  to  be  a  par- 
ticular case  of  a  well  known  principle  of  arithmetical  proportions  ; 
the  following  is  a  more  complete  proof  of  it.  Let  Z  (Fig.  31)  rep- 
resent the  zenith,  H  O  R  the  horizon,  P  the  pole,  and  S,  S'  the 
points  at  which  the  upper  and  lower  transits  of  a  circumpolar  star 
take  place ;  H  P  will  be  equal  to  the  latitude,  and  Z  P  will  be 
equal  to  the  co-latitude.     Now,  we  have, 

H  P  =  H  S  +  P  S,  and  H  P  -  H  S'  —  P  S'  =  H  S'  —  P  S 

whence,  2  HP  =  H  S  +  H  S,  or,  H  P  =  ^  ^  +  ^  ^' .  .  .  (24). 

In  like  manner  we  obtain, 

Z  P  =  ^  ^  +  ^  ^'  .  .  .  (25). 

Wherefore,  let  the  altitudes  of  a  circumpolar  star  at  its  upper  and 
lower  transits,  be  measured  and  corrected  for  refraction,  and  their 
half  sum  will  be  the  latitude ;  or,  let  the  zenith  distances  be 
measured,  and  corrected  for  refraction,  and  their  half  sum  sub- 
tracted from  90°  will  be  the  latitude.  Stars  should  be  selected 
that  have  a  considerable  altitude  at  their  inferior  transit,  for,  the 
greater  is  the  altitude  the  less  is  the  uncertainty  as  to  the  amount 
of  the  refraction.  On  this  principle  the  pole  star  is  to  be  preferred 
to  all  others. 

2.  By  a  single  meridian  altitude  or  zenith  distance.  Let  s, 
s',  s"  (Fig.  6)  be  the  points  of  meridian  passage  of  three  different 
stars,  the  first  to  the  north  of  the  zenith,  the  second  between  the 
zenith  and  equator,  and  the  third  to  the  south  of  the  equator :  Z  E 
=  the  latitude,  and  we  have  for  the  three  stars, 

Z  E  =  s E  —  Z  5,  Z E  =  5'  E  4-  Z  5,  Z  E  =  Z 5"  —  5"  E. 


LATITUDE  AND  LONGITUDE  OF  A  PLACE.         65 

Thus,  if  the  zenith  distance  be  called  north  or  south  according  as 
the  zenith  is  north  or  south  of  the  star  when  on  the  meridian,  in 
case  the  zenith  distance  and  declination  are  of  the  same  name 
their  sum  will  be  equal  to  the  latitude ;  but  if  they  are  of  different 
names,  their  difference  will  be  the  latitude,  of  the  same  name  with 
the  greater. 

Tliis  method  supposes  the  declination  of  the  body  to  be  known. 
The  declination  of  a  star  or  of  the  sun  at  any  time  is,  in  practice, 
obtained  for  the  solution  of  this  and  other  problems,  by  the  aid  of 
tables,  or  is  taken  by  inspection  from  the  English  Nautical  Alma- 
nac or  other  similar  work.  If  the  time  of  the  meridian  transit  be 
known,  the  altitude  may  be  measured  by  a  sextant  (Art.  66). 
The  observed  altitude  must  be  corrected  for  refraction,  and  also 
for  parallax  if  the  body  observed  is  the  sun,  or  moon,  or  either  one 
of  the  planets. 

This  method  of  finding  the  latitude  is  the  one  most  generally 
employed  at  sea,  the  sun  being  the  object  observed.  As  the  time 
of  noon  is  not  known  with  accuracy,  several  altitudes  about  the 
time  of  noon  are  taken,  and  the  meridian  altitude  is  deduced  from 
these. 

151.  The  astronomical  latitude  being  known,  the  reduced  lati- 
tude (p.  15,  def  4)  may  be  obtained  by  subtracting  from  it  the 
reduction  of  latitude.  For,  if  O  C  (Fig.  30)  represents  the  radius 
and  O  N  the  vertical,  at  any  place  O,  and  E  C  Q,  represents  the 
terrestrial  equator,  O  N  Q  will  be  the  astronomical  latitude,  O  C  Q. 
the  reduced  latitude,  and  CON  the  reduction  of  latitude ;  and 
we  have 

ONGl  =  OCa  +  CO  N,andOCa  =  ONa— CON  ..(26). 
(For  the  practical  method  of  resolving  this  problem,  see  Prob.  XV). 

152.  There  are.  various  methods  of  findinsf  the  lonsfitude  of  a 
place,  nearly  all  of  which  rest  upon  the  following  principle  : 

The  difference  at  any  instant  between  the  local  times^  {whether 
sidereal  or  solar),  at  any  -place  and  on  the  first  meridian,  is  the 
longittide  of  the  place,  expressed  in  time  ;  and  consequently,  also, 
the  difference  between  the  local  times  at  any  two  places,  is  their 
difference  of  longitude,  in  time. 

The  truth  of  this  principle  is  easily  established.  In  the  first 
place,  we  remark  that  the  longitude  of  a  place  contains  the  same 
number  of  degrees  and  parts  of  a  degree  as  the  arc  of  the  celes- 
9 


66  Astronomy. 

tial  equator  comprised  between  the  meridian  of  Greenwich  and 
the  meridian  of  the  place.  Now,  it  is  Oh.  Om.  Os.  of  mean 
solar  time,  or  mean  noon  at  any  place,  when  the  mean  snn  (Art. 
39)  is  on  the  meridian  of  that  particular  place.  Therefore,  as 
the  mean  sun,  moving  in  the  equator,  recedes  from  the  meridian 
towards  the  west  at  the  rate  of  15°  per  mean  solar  hour,  when  it 
is  mean  noon  at  a  place  to  the  west  of  Greenwich,  it  will  be 
as  many  hours  and  parts  of  an  hour  j)ast  mean  noon  at  Green- 
wich, as  is  expressed  by  the  quotient  of  the  division  of  the  arc 
of  the  celestial  equator,  or  its  equal  the  longitude,  by  15.  If  the 
place  be  to  the  east^  instead  of  to  the  west  of  Greenwich,  when 
it  is  mean  noon  there,  it  will  be  as  much  before  mean  noon  at 
Greenwich  as  is  expressed  by  the  longitude  of  the  place  con- 
verted into  time  (as  above).  In  either  situation  of  the  place, 
then,  the  principle  just  stated  will  be  true. 

It  is  plain  that  the  equality  between  the  differences  of  the 
times  and  of  the  longitudes  will  subsist  equally,  if  sidereal 
instead  of  solar  time  be  used. 

153.    To  find  the  longitude  of  a  place. 

1.  Let  two  observers,  stationed  one  at  Greenwich,  and  the  other 
at  the  given  place,  note  tJie  times  of  the  occurrence  of  so7ne  phe- 
nomenon which  is  seen  at  the  same  instant  at  both  places  ;  the 
difference  of  the  observed  times  will  be  the  longitude  in  time. 
These  same  observations  made  at  any  two  places  will  make 
known  their  difference  of  longitude.  If  the  stations  are  not  dis- 
tant from  each  other,  a  signal,  as  the  flashing  of  gunpowder,  or 
the  firing  of  a  rocket,  may  be  observed.  When  they  are  remote 
from  each  other,  celestial  phenomena  must  be  taken.  Eclipses 
of  the  satellites  of  Jupiter  and  of  the  moon  are  phenonema 
adapted  to  the  purpose  in  question.  However,  as  in  these 
eclipses  the  diminution  of  the  light  of  the  body  is  not  sudden,  but 
gradual,  the  longitude  cannot  be  obtained  with  very  great  accu- 
racy from  observations  made  upon  them. 

2.  Transport  a  chronometer  which  has  beeti  carefully  adjusted 
to  the  local  time  at  Greenwich,  to  the  place  whose  longitude  is 
sought,  and  compare  the  time  given  by  the  chronometer  with  the 
local  time  of  the  place.  In  the  same  way,  by  transporting  a 
chronometer  from  any  one  place  to  another,  their  difference 
of  longitude   may   be   obtained.     The   error  and  rate  of  the 


PLACES    OF    THE    FIXED    STARS.  G7 

chronometer  must  be  determined  at  the  outset,  and  as  often  after- 
wards as  circumstances  will  admit,  that  the  error  at  the  moment 
of  the  observation  may  be  known  as  accurately  as  possible.  To 
insure  greater  certainty  and  precision  in  the  knowledge  of  the 
time,  three  or  four  chronometers  are  often  taken,  instead  of  one 
only. 

This  method  is  much  used  at  sea ;  the  local  time  being  ob- 
tained from  an  observation  upon  the  sun  or  some  other  heavenly 
body,  in  a  manner  to  be  hereafter  explained. 

3.  Let  the  Greenwich  time  of  the  occurrence  of  some  celestial 
phenomeno?i  be  computed,  and  note  the  time  of  its  occurrence  at 
the  given  place. 

Eclipses  of  the  sun  and  moon,  and  of  Jupiter's  satellites,  oc- 
cultations  of  the  stars  by  the  moon,  and  the  angular  distance  of 
the  moon  from  some  one  of  the  heavenly  bodies,  are  the  phenome- 
na employed.  The  Greenwich  times  of  the  beginning  and  end  of 
the  eclipses  of  Jupiter's  satellites,  are  published  for  the  solution 
of  the  problem  of  the  longitude,  in  the  English  Nautical  Alma- 
nac. Eclipses  of  the  sun  and  occultations  of  the  stars  furnish 
the  most  exact  determinations  of  the  longitude,  but  they  cannot 
be  used  for  this  purpose  unless  the  longitude  is  already  approxi- 
mately known. 

The  explanation,  in  detail,  of  the  method  of  lunar  distances, 
which  is  chiefly  used  at  sea,  may  be  found  in  treatises  on  Naviga- 
tion and  Nautical  Astronomy. 


CHAPTER    VI. 

OF    THE    PLACES    OF    THE    FIXED    STARS. 

154.  The  place  of  a  fixed  star  in  the  sphere  of  the  heavens,  is 
found  by  ascertaining  its  true  right  ascension  and  declination, 
which  are  the  co-ordinates  of  its  place.  The  process  of  finding 
the  true  right  ascension  and  declination  of  a  heavenly  body  has 


08  ASTRONOMY. 

already  been  detailed :  the  apparent  right  ascension  and  declina- 
tion are  found  as  explained  in  Arts.  48,  59,  and  to  these  are 
applied  the  several  corrections  of  refraction,  parallax  (when  sen- 
sible), and  aberration  (Arts.  75,  106,  114). 

When  riofht  ascensions  and  declinations  found  at  different 
times  are  to  be  compared  together,  or  employed  in  the  same  cal- 
culations, as  often  becomes  necessary,  they  are  to  be  reduced  to 
the  same  epoch  by  correcting  for  precession  and  nutation  (p.  57). 

155.  It  is  important  to  observe,  however,  that  the  places  of  the 
fixed  stars,  as  at  present  known,  were  not  obtained  by  the  direct 
process  just  referred  to,  that  is,  by  observing  the  right  ascension  and 
declination,  and  applying  to  them  at  once  all  the  corrections  of 
which  we  have  treated.  They  were  arrived  at  by  successive 
approximations.  The  respective  corrections  were  applied  in 
succession  as  they  came  to  be  discovered  ;  and  more  accu- 
rate results  were  obtained,  as,  by  the  improvement  of  the  instru- 
ments, the  observations  became  more  and  more  exact,  and  as  the 
amount  of  the  corrections  came  to  be  known  with  greater  and 
greater  precision. 

156.  In  order  to  distinguish  the  fixed  stars  from  each  other, 
they  are  arranged  into  groups,  called  Constellatiojis,  which  are 
imagined  to  form  the  outlines  of  figures  of  men,  animals,  or  other 
objects,  from  which  they  are  named.  Thus  one  group  is  con- 
ceived to  form  the  figure  of  a  Bear,  another  of  a  Lion,  a  third  of 
a  Dragon,  and  a  fourth  of  a  Lyre.  The  division  of  the  stars  into 
constellations  is  of  very  remote  antiquity ;  and  the  names  given 
by  the  ancients  to  individual  constellations  are  still  retained. 

The  constellations  are  divided  into  three  classes :  Northern 
Constellations,  Southei^n  Constellations,  and  Constellations  of 
the  Zodiac.  Their  whole  number  is  91 :  Northern  34,  Southern 
45,  and  Zodiacal  12.  The  number  of  the  ancient  constellations 
was  but  48.  The  rest  have  been  formed  by  modern  astrono- 
mers, from  southern  stars  not  visible  to  the  ancient  observers, 
and  others  variously  situated,  which  escaped  their  notice,  or 
were  not  attentively  observed. 

157.  The  stars  of  a  constellation  are  distinguished  from  each 
other  by  the  letters  of  the  Greek  alphabet,  and  in  addition  to 
these,  if  necessary,  the  Roman  letters,  and  the  numbers  1,  2,  3, 
(fcc. ;  the  characters,  according  to  their  order,  denoting  the  rela- 


PLACES    OF    THE    FIXED    STARS.  G9 

live  mao-nitudes  of  the  stars.  Thus,  a  Arietis  designates  the 
largest  star  in  the  constellation  Aries ;  (3  Draconis,  the  second 
star  of  the  Dragon,  &c. 

Some  of  the  fixed  stars  have  particular  names,  as  Sirius, 
Aldebaran,  Arcturus,  Reguhis,  &c. 

158.  The  stars  are  also  divided  into  classes,  or  magnitudes, 
accordmg  to  the  degrees  of  their  apparent  brightness.  The 
largest  or  brightest  are  said  to  be  of  the  Jiist  magnitude  ;  the 
next  in  order  of  brightness,  of  the  seco7id  magnitude  ;  and  so 
on  to  stars  of  the  sixth  magnitude,  which  includes  all  those  that 
are  barely  perceptible  to  the  naked  eye.  All  of  a  smaller  kind 
are  called  telescopic  stars,  being  invisible  without  the  assistance 
of  the  telescope.  The  classification  according  to  apparent  mag- 
nitude is  continued  with  the  telescopic  stars  down  to  stars  of  the 
sixteenth  magnitude. 

159.  The  places  of  the  fixed  stars  are  generally  expressed 
by  their  right  ascensions  and  declinations,  but  sometimes  also 
by  their  longitudes  and  latitudes.  A  table  containing  a  list  of 
fixed  stars,  designated  by  their  proper  characters,  and  giving 
their  mean  right  ascensions  and  declinations,  or  their  mean  lon- 
gitudes and  latitudes,  is  called  a  Catalogue  of  those  stars.* 

Table  I^I  is  a  catalogue  of  fifty  principal  fixed  stars,  and  gives  yC(^ , 
their  mean  right  ascensions  and  declinations  for  the  beginning 
of  the  year  1840,  as  well  as  their  annual  variations  in  right 
ascension  and  declination.  The  annual  variations  serve  to  ex- 
tend the  use  of  the  catalogue  about  10  years  (Art.  135)  before  and 
after  the  epoch  for  which  it  is  constructed.  (See  Prob.  XVIII.) 
Every  ten  years,  or  thereabouts,  a  new  catalogue  must  be  formed. 

160.  If  the  true  right  ascension  and  declination  of  a  star  at  a 
given  time  be  required,  correct  the  mean  right  ascension  and 
declination  found  by  the  catalogue,  for  nutation.  (See  Art.  132.) 
And  if  the  apparent  right  ascension  and  declination  be  re- 
quired, correct  also  for  aberration.     (See  Art.  114.) 

161.  The  latitude  and  longitude  of  a  fixed  star  or  other  hea- 


*  Various  catalogues  have  at  different  periods  been  published.  The  first  cata- 
logue was  begun  by  Hipparchus  120  years  before  the  Christian  era.  The  most 
modern  and  most  accurate  catalogues,  although  not  the  most  extensive,  are  the 
catalogues  of  Lacaille,  Bradley,  Mayer,  and  Maskelyne. 


70  ASTRONOMY. 

venly  body  are  obtained  originally  by  computation  from  its  right 
ascension  and  declination. 

To  convert  the  right  ascension  and  declination  of  a  body  into 
its  longitude  and  latitude. 

Let  E  Q,  (Fig.  32)  represent  the  equator,  E  C  the  echptic,  P,  K 
the  poles  of  the  equator  and  ecliptic,  E  the  vernal  equmox,  P  S 
R  a  circle  of  declination,  and  K  S  L  a  circle  of  latitude,  both 
passing  through  a  body  S.  The  right  ascension  of  the  body  is 
E  R  =  R ;  the  declination  R  S  =  D  ;  the  longitude  E  L  =  L  ;  and 
the  latitude  LS=X.  REL  =  wis  the  obliquity  of  the  ecliptic, 
which  is  one  of  the  essential  data  of  the  problem.  R  E  S  =  ar  and 
L  E  S  =  y  are  employed  as  auxiliary  angles.  In  the  right  angled 
triangle  L  E  S,  we  have  by  Napier's  rules  for  the  solution  of  right 
angled  triangles, 

sin  (co.  L  E  S)  =  tang  E  L  tang  (co.  E  S) ; 
whence, 

tan  E L  =  cos  L E S  tan  E S,  or, tan  L  =  cos  (RES  —  w)  tan  E  S  ; 
but, 

sin  (CO.  R  E  S)  =  tan  E  R  tan  (co.  E  S),  or,  tan  E  S  =  ^'^"^  ^  ^ ; 
^  ^  ^  ^  cos  RES' 

thus, 

T  /Tj  -r-  o        \  tang  E  R      cos  (x  —  w)  tan  R        ..^^^ 

tan  L  =  cos  (R  E  S  —  u)  — °,  ^  ^  = ^ ^- . .  (27). 

cos  R  E  S  cos  X 

And  to  find  x,  we  have 

sin  E  R  =  tan  (co.  RES)  tan  R  S,  or,  cot  a:  =  sin  R  cot  D  .  .  (28). 

Again, 

sin  E  L  =  tan  (co.  L  E  S)  tan  L  S,  or  tan  L  S  =  tan  L  E  S  sin  E  L, 

which  gives,  tang  X  =  tang  {x  —  w)  sin  L  .  .  .  (29). 

Equation  (28)  makes  known  the  value  of  x,  with  which  we 
derive  the  values  of  L  and  X,  by  means  of  equations  (27)  and  (29). 
In  resolving  the  equations,  attention  must  be  paid  to  the  signs 
of  the  quantities,  which  are  determined  according  to  the  usual 
trigonometrical  rules,  it  beins:  understood  that  the  declination  D 
is  to  be  regarded  as  negative  when  it  is  south,  x  is  to  be  taken 
always  less  than  180°,  and  greater  or  less  than  90°  according 
as  its  cotangent  is  negative  or  positive.  L  will  always  be  in  the 
same  quadrant  with  R.  The  latitude  X  will  be  north  or  south, 
according  as  tang  X  comes  out  positive  or  negative. 

The  apparent  or  mean  obliquity  is  used,  according  as  the  case 


PLACES     OF    THE    FIXED    STARS.  "'Ttl  ' 

refers  to  true  or  mean  co-ordinates.     (For  exemplifications  of 
this  problem,  see  Prob.  XXIV.) 

162.  It  is  now  frequently  necessary  to  resolve  the  converse 
problem,  that  is,  to  convert  the  longitude  and  latitude  of  a  body 
into  its  right  ascension  atid  declination. 

The  triangle  RES  (Fig  32)  gives, 

sin  (CO.  R  E  S)  =  tang  E  R  tang  (co  E  S) ; 
whence, 

tan  E  R  =  COS  R  E  S  tan  E  S,  or,  tan  R  =  cos  (L  E  S  +  w)  tan  E  S  ; 
but, 

sin  (CO.  L  E  S)  =  tang  E  L  tang  (co.  E  S),  or  tan  E  S  =  ^^M^-i^ ; 

cos  L  E  S 

thus, 

tang  R  =  cos  (L  E  S  +  c.)  !^^  =  cos  (y  +  c)  tan^L    ,3^ 

cosLES  cos  y 

and  to  find  y,  we  have 

sin  E  L  =  tang  (co.  L  E  S)  tang  L  S,  or  cot  y  =  sin  L  cot  X  .  (31). 
For  the  declination,  we  have 

sin  E  R=tan  (co.  RES)  tan  R  S,  or,  tan  R  S  =  tan  R  E  S  sin  E  R ; 
or,  tang  D  =  tang  (y  -f  u)  sin  R     ...  (32). 

The  value  of  y  being  derived  from  equation  (31),  and  substi- 
tuted in  equations  (30)  and  (32),  these  equations  will  then  make 
known  the  values  of  R  and  D.  The  signs  of  the  quantities  are 
determined  by  the  usual  trigonometrical  rules,  the  latitude  X 
being  taken  negative  when  south,  y  is  always  less  than  180°, 
and  greater  or  less  than  90°  according  as  its  cotangfent  comes 
out  negative  or  positive.  R  will  be  in  the  same  quadrant  as  L. 
The  declination  will  be  north  or  south,  according  as  its  tangent 
comes  out  positive  or  negative.  (For  exemplifications  of  this 
problem,  see  Prob.  XXV.) 

163.  Table  XCII  contains  the  mean  longitudes  and  latitudes 
of  some  of  the  principal  fixed  stars  for  the  beginnmg  of  the  year 
1840,  together  with  their  annual  variations  which  serve  to 
make  known  the  mean  longitudes  and  latitudes  at  any  other 
epoch.     (See  Prob.  XVIII.) 

164.  There  are  two  principal  modes  of  representing  the 
stars ;  the  one  by  delineating  them  on  a  globe,  where  each  star 
occupies  the  spot  in  which  it  would  appear  to  an  eye  placed  in 
the  centre  of  the  sflobe,  and  where  the  situations  are  reversed 


72  ASTRONOMY. 

when  we  look  down  upon  them  ;  the  other  mode  is  by  a  chart, 
where  the  stars  are  generally  so  arranged  as  to  represent  them 
in  positions  similar  to  their  natural  ones,  or  as  they  would  appear 
on  the  internal  concave  surface  of  the  globe.  The  construction 
of  "a  globe  or  chart  is  effected  by  means  of  the  right  ascensions 
and  declinations  of  the  stars.  The  point  which  represents  the 
place  of  a  star  is  found  by  marking  off  the  rijjht  ascension  and 
declination  of  the  star,  upon  the  globe. 

All  the  fixed  stars  visible  to  the  naked  eye,  together  with  some 
of  the  telescopic  stars,  are  represented  on  celestial  globes  of  12  or 
18  inches  in  diameter. 

165.  The  fixed  stars  (so  called)  are  not  all  of  them,  rigorously 
speaking,  fixed  or  stationary  in  the  heavens.  It  has  been  dis- 
covered that  many  of  them  have  a  very  slow  motion  from  year 
to  year.  These  motions  of  the  stars  are  called  Proper  Motiotis. 
The  annual  variations  in  right  ascension  and  declination,  and  in 
longitude  and  latitude,  given  in  Tables  XC  and  XCII,  are  the 
variations  due  both  to  the  precession  of  the  equinoxes  and  the 
proper  motions  of  the  stars. 


CHAPTER    VII. 


OF    THE    APPARENT  MOTION    OF    THE    SUN    IN    THE    HEAVENS, 

166.  The  sun's  declination,  and  the  diflference  of  right  ascen- 
sion of  the  sun  and  some  fixed  star,  found  from  day  to  day 
throughout  a  revolution,  are  the  elements  from  which  the  cir- 
cumstances of  the  sun's  apparent  motion  are  derived. 

The  motion  of  the  sun  as  at  present  known,  has  been  arrived 
at  in  the  same  approximative  manner  as  the  places  of  the  fixed 
stars  (Art.  155).  It  would  in  fact  be  theoretically  impossible  to 
correct  the  co-ordinates  of  the  sun's  apparent  place  for  precession, 
nutation,  and  aberration,  in  the  original  determination  of  the 


OBLiaUITY    OF    THE    ECLIPTIC.  73 

sun's  motion  ;  for,  the  knowledge  of  these  corrections  pre-sup- 
poses  some  knowledge  of  the  motion  of  the  sun. 

167.  The  curve  on  the  sphere  of  the  heavens  passing  through 
the  successive  positions  determined  as  above  from  day  to  day,  is 
the  ecliptic.  If  we  suppose  it  to  be  a  circle,  as  it  appears  to  be, 
its  position  will  result  from  the  position  of  the  equinoctial  points, 
and  its  obliquity  to  the  equator. 

168.  To  find  the  ohUquity  of  the  ecliptic. 

Let  E  Gt  A  (Fig.  33)  represent  the  equator,  EGA  the  ecliptic, 
and  O  C,  O  E  lines  drawn  through  O  the  centre  of  the  earth  and 
perpendicular  to  O  E  the  line  of  the  equinoxes  ;  then  the  angle 
C  O  d  will  be  the  obliquity  of  the  ecliptic.  This  angle  has  for 
its  measure  the  arc  C  Q,  and  therefore  the  ohliquity  of  the  eclip- 
tic is  equal  to  the  greatest  declination  of  the  sun.  It  can  but 
rarely  happen  that  the  time  of  the  greatest  declination  will  coin- 
cide with  the  instant  of  noon  at  the  place  where  the  observations 
are  made,  but  it  must  fall  within  at  least  twelve  hours  of  the  noon 
for  which  the  observed  declination  is  the  greatest.  In  this 
interval  the  change  of  declination  cannot  exceed  4",  and  there- 
fore the  greatest  observed  declination  cannot  differ  more  than  4" 
from  the  obliquity.  A  formula  has  been  investigated,  which 
gives  in  terms  of  determinable  quantities,  the  difference 
between  any  of  the  greater  declinations  and  the  maximum 
declination.  By  reducing  by  means  of  this  formula  a  num- 
ber of  the  greater  declinations  to  the  maximum  declination, 
and  taking  the  mean  of  the  individual  results,  a  very  accurate 
value  of  the  obliquity  may  be  found. 

169.  To  find  the  position  of  the  vernal  or  autumnal  equi- 
no.v. 

1.  On  inspecting  the  observed  declinations  of  the  sun,  it  is 
seen  that  about  the  21st  of  March  the  declination  changes  in  the 
interval  of  two  successive  noons  from  south  to  north.  The 
vernal  equinox  occurs  at  some  moment  of  this  interval.  Let 
R  S,  R'  S'  (Fig.  34)  represent  the  declinations  at  the  noons  be- 
tween which  the  equinox  occurs  :  as  one  is  north  and  the  other 
south,  their  sum  (S)  will  be  the  daily  change  of  declination  at 
the  time  of  the  equinox.  Denote  the  time  from  noon  to  noon 
by  T.  Now,  to  find  the  interval  {x)  between  the  noon  preced- 
10 


/4  ASTRONOMY. 

ing  the  equinox  and  the  instant  of  the  equinox,  state  the  pro- 
portion 

o 
Next,  take  the  daily  change  in  right  ascension  (R  R')  on  the  day 
of  the  equinox  and  compute  the  value  of  R  E,  by  the  pro- 
portion 

T::r,or  Z^^  : :  R  R' :  RE  ; 

S 

on  the  principle  that  the  declination  changes  for  a  day  or  more 
proportionally  to  the  time.  Add  R  E  to  M  R  the  observed  dif- 
ference of  right  ascension  (Art.  166)  on  the  day  preceding  the 
equinox,  and  the  sum  M  E  will  be  the  distance  of  the  equinox 
from  the  meridian  of  the  star  observed  in  connection  with 
the  sun.* 

The  position  of  the  autumnal  equinox  may  be  found  by  a 
similar  process,  the  only  difference  in  the  circumstances  being 
that  the  declination  chanofes  from  north  to  south  instead  of  from 
south  to  north. 

If  the  value  of  x  which  results  from  the  first  proportion,  be 
added  to  the  time  of  noon  on  the  day  preceding  the  equinox, 
the  result  will  be  the  time  of  the  equinox. 

2.  In  the  triangle  RES  (Fig.  33)  we  have  the  angle  R  E  S  =  u 
the  obliquity  of  the  ecliptic,  and  R  S  =  D  the  declination  of  the 
sun,  both  of  which  we  may  suppose  to  be  known,  and  we  have 
by  Napier's  rules, 

sin  E  R  =  tang  (co.  RES)  tang  R  S  =  cot  u  tang  D  .  .  (33) ; 
whence  we  can  find  E  R.  And,  by  taking  the  sum  or  difference 
of  E  R  and  M  R,  according  as  the  star  is  on  the  opposite  side  of 
the  sun  from  the  equinox,  or  the  same  side,  we  obtain  M  E  as 
before.  If  this  calculation  be  effected  for  a  number  of  positions 
S,  S'j  S",  &.C.  of  the  sun  on  different  days,  and  a  mean  of  all  the 
individual  results  be  taken,  a  more  exact  value  of  M  E  will  be 
obtained. 

M  E  being  accurately  known,  the  precise  time  of  the  equinox 
may  readily  be  deduced  from  the  observed  daily  variation  of 
right  ascension  on  the  day  of  the  equinox. 

*  The  star  is  here  supposed  to  be  to  tlie  west  of  the  sun. 


DETERMINATION    OP    THE    LONGITUDE    OP    THE    SUN.  75 

170.  The  calculations  just  mentioned  rest  upon  the  hypo- 
thesis that  the  ecliptic  is  a  great  circle.  The  close  agreement 
which  is  found  to  subsist  between  the  values  of  M  E  deduced 
from  observations  upon  the  sun  in  different  positions  S,  S',  S", 
(fcc,  establishes  the  truth  of  this  hypothesis.  It  is  also  con- 
firmed by  the  fact,  that  the  right  ascensions  of  the  vernal  and 
autumnal  equinox  differ  by  180°,  since  we  may  infer  from  this 
that  the  line  of  the  equinoxes  passes  through  the  centre  of 
the  earth. 

171.  The  mean  obliquity  of  the  ecliptic  is  derived  from  the 
apparent  obliquity,  as  well  as  the  mean  equinox  from  the  true 
equinox,  by  correcting  for  nutation. 

172.  The  mean  obliquity  at  any  one  epoch  having  been 
found,  its  value  at  any  assumed  time  may  be  deduced  from  it  by 
allowing  for  the  annual  diminution  of  0".46  (see  Table  XXII). 
In  like  manner,  the  place  of  the  mean  equinox  at  any  given  time 
may  be  derived  from  its  place  once  found,  by  allowing  for  the 
annual  precession  of  50".2. 

The  mean  obliquity  having  thus  been  found  for  any  assumed 
time,  the  apparent  obliquity  at  the  same  time  becomes  known, 
by  applying  the  nutation  of  obliquity.     (See  Prob.  X.) 

173.  The  longitude  of  the  sun  may  be  expressed  in  terms  of 
the  obliquity  of  the  ecliptic  and  the  right  ascension  or  declina- 
tion. In  the  triangle  ERS,  (Fig.  33)  E  S  (=  L)  represents  the  lon- 
gitude of  the  sun  supposed  to  be  at  S,  E  R  (—  R)  its  right  ascen- 
sion, and  R  S  (=  D)  its  declination.     Now,  by  Napier's  rules, 

cos  R  E  S  =  tang  E  R  cot  E  S,  or  cot  E  S  =  ^"^  ^  ^  ^  = 
"=  tang  E  R 

cos  R  E  S  cot  E  R ; 

thus, 

cot  L  =  cos  w  cot  R,  or  tang  L  =  — => —  .  .  .  (34). 

cos  w 

Also, 

sinRS 


sin  R  E  S  ' 


sinRS=cos(co.RES)cos(co.ES);  whence,  sinES  = 

.    T       sin  D  /Of', 

or,  sm  L  =  - —   .  .  .  (35). 

sin  w 

With  these  formulae  the  longitude  of  the  sun  may  be  computed 

from  either  its  right  ascension  or  declination.    (See  Prob.  XII.) 


76  ASTRONOMY. 

Formulas  (34)  and  (35)  may  be  written  thus, 

tang  R  =  tang  L  cos  w;  sin  D  =  sin  L  sin  w  .  .  .  (36). 

These  formulae  will  make  known  the  right  ascension  and  de- 
clination of  the  sun,  when  his  longitude  is  given.  (See  Prob. 
XI.)  It  will  be  seen  in  the  sequel,  that  in  the  present  advanced 
state  of  astronomical  science,  the  longitude  of  the  sun  at  any 
assumed  time  may  be  computed  from  the  ascertained  laws  and 
rate  of  the  sun's  motion. 

174.  The  interval  between  two  successive  returns  of  the  sun 
to  the  same  equinox,  or  to  the  same  longitude,  is  called  a  Tro- 
pical  Year. 

And  the  interval  between  two  successive  returns  of  the  sun  to 
the  same  position  with  respect  to  the  fixed  stars,  is  called  a  Si- 
dereal Year. 

175.  It  appears  from  observation  that  the  length  of  the  tropi- 
cal year  is  subject  to  slight  periodical  variations.  The  period 
from  which  it  deviates  periodically  and  equally  on  both  sides,  is 
called  the  Mean  Tropical  Year.  As  the  changes  in  the  length 
of  the  true  tropical  year  are  very  minute,  the  length  of  the  mean 
tropical  year  is  obviously  very  nearly  equal  to  the  mean  length 
of  the  true  tropical  year,  in  an  interval  during  which  it  passes 
one  or  more  times  through  all  its  different  values.  In  point  of 
fact,  it  may  be  found  with  a  very  close  approximation  to  the 
truth,  by  comparing  two  equinoxes  observed  at  an  interval  of  60 
or  100  years. 

Theory  shows  that  the  variation  in  the  length  of  the  tropical 
year  arises  from  the  periodical  inequality  in  the  precession  of 
the  equinoxes  which  results  from  nutation,  and  certain  periodi- 
cal inequalities  in  the  sun's  yearly  rate  of  motion  ;  and  thus 
establishes  also,  that  the  mean  tropical  year,  as  above  defined,  is 
the  same  as  the  interval  between  two  successive  returns  of  the 
sun,  supposed  to  have  its  mean  motion,  to  the  same  mean  equi- 
nox. According  to  the  most  accurate  determinations,  the  length 
of  the  mean  tropical  year,  expressed  in  mean  solar  time,  is  365  d. 
5h.  48  m.  47.58 s. 

176.  In  a  mean  tropical  year  the  sun's  mean  motion  in  longi- 
tude is  360°  ;  hence,  to  find  his  mean  daily  motion  in  longitude, 
we  have  only  to  state  the  proportion 

365d.  5h.  48m.  47s. :  Id. ;  :  360°  •.x  =  m'  8".33. 


Kepler's  laws.  77 

177.  The  sidereal  year  is  lojiger  than  the  tropical.  For, 
since  the  equinox  has  a  retrograde  motion  of  50". 23  in  a  year, 
when  the  sun  has  returned  to  the  equinox  it  will  not  have  ac- 
complished a  sidereal  revolution,  into  50".23.  The  excess  of 
the  sidereal  over  the  tropical  year  results  from  the  proportion 

59'  8  ".3  :  50".23  :  :  Id.  :  :r  =  20m.  23.1s. 
Thus  the  length  of  the  mean  sidereal  year,  expressed  in  mean 
solar  time,  is  365d.  6h.  9m.  10.7s. 

178.  If  from  the  right  ascensions  and  declinations  of  the  sun, 
found  on  two  successive  days,  the  corresponding  longitudes  be 
deduced  (equa.  34,  35),  and  their  difference  taken,  the  result 
will  be  the  sun's  daily  motion  in  longitude  at  the  time  of  the 
observations.  The  sun's  daily  motion  in  longitude  is  not  the 
same  throughout  the  year,  but,  on  the  contrary,  is  continually 
varying.  It  gradually  increases  during  one  half  of  a  revolution, 
and  gradually  decreases  during  the  other  half,  and  at  the  end  of 
the  year  has  recovered  its  original  value.  Thus,  the  greatest 
and  least  daily  motions  occur  at  opposite  points  of  the  ecliptic. 
They  are,  respectively,  61'  10"  and  57'  11". 

179.  The  exact  laws  of  the  sun's  unequable  motion  cannot  be 
obtained,  without  a  knowledge  of  the  law  of  variation  of  the 
sun's  distance,  or,  what  amounts  to  the  same,  of  the  form  of  the 
orbit  apparently  described  by  the  sun  in  space. 


CHAPTER    VIII. 

OF    THE    MOTIONS    OF    THE   SUN,    MOON,    AND   PLANETS, 
IN    THEIR    ORBITS. 

Kepler's  Laws. 

180.  The  celebrated  astronomer,  Kepler,  discovered  from  ob- 
servation, that  the  motions  of  the  planets,  including  the  earth, 
are  in  conformity  with  the  three  following  laws  : 

1.  The  areas  described  by  the  radius  vector  of  a  planet  [or 


ASTRONOMY. 


the  line  drawn  from  the  «ui.  to  the  planet]  are  proportional  to 
the  times. 

2.  The  orbit  of  a  planet  is  an  ellipse,  of  which  the  sun  occu- 
pies one  of  the  foci. 

3.  The  squares  of  the  times  of  revolution  of  the  planets  are 
proportional  to  the  cubes  of  their  mean  distances  from  the  sun, 
or  of  the  semi-major  axes  of  their  orbits. 

These  laws  are  known  by  the  denomination  oi  Kepler's  Laivs. 
They  were  first  announced  by  Kepler  as  the  fundamental  laws  of 
the  planetary  motions,  from  a  partial  examination  only  of  these 
motions.  The  first  two  he  assumed  as  hypotheses,  after  that 
he  had  discovered  that  the  radius  vector  and  angular  motion  of 
a  planet  were  variable,  and  afterwards  verified  them,  or  rather 
partiaUy  verified  them.  They  have  since  been  completely 
verified  by  other  astronomers.  We  shall  accordingly  adopt 
these  two  laws  for  the  present  as  hypotheses,  and  show  in  the 
sequel  that  they  are  verified  by  the  results  deducible  from 
them. 

These  laws  being  established,  the  third  is  obtained  by  simply 
comparing  the  known  major  axes  and  times  of  revolution. 

181.  The  apparent  motion  of  the  sun  in  space  must  be  sub- 
ject to  Kepler's  first  two  laws  ;  for,  the  apparent  orbit  of  the  sun 
is  of  the  same  form  and  dimensions  as  the  actual  orbit  of  the 
earth,  and  the  law  and  rate  of  the  sun's  motion  in  its  apparent 
orbit,  are  the  same  as  the  law  and  rate  of  the  earth's  motion. 
To  establish  these  two  facts,  let  E  E'  A  (Fig.  35)  represent  the 
elliptic  orbit  of  the  earth,  and  S  the  position  of  the  sun  in  space. 
If  the  earth  move  from  E  to  any  point  E',  as  it  seems  to  remain 
stationary  at  E,  it  is  plain  that  the  sun  will  appear  to  move  from 
S  to  a  position  S',  on  the  line  E  S'  drawn  parallel  to  E'  S  the 
actual  direction  of  the  sun  from  the  earth,  and  at  a  distance  E 
S'  equal  to  E'  S  the  actual  distance  of  the  sun  from  the  earth. 
Thus,  for  every  position  of  the  earth  in  its  orbit,  the  correspond- 
ing apparent  position  of  the  sun  is  obtained  by  drawing  a  line 
parallel  to  the  radius  vector  of  the  earth,  and  equal  to  it.  It 
follows,  therefore,  that  the  area  S  E  S'  apparently  described  by 
the  radius  vector  of  the  sun  (or  the  line  drawn  from  the  sun  to 
the  earth)  in  any  interval  of  time,  is  equal  to  the  area  E  S  E' 
actually  described  by  the  radius  vector  of  the  earth  in  the  same 


ANGULAR    MOTION    OF    A    PLANET.  79 

time  ;  and,  consequently,  that  the  arc  S  S'  apparently  described 
by  the  sun  in  space,  is  equal  to  the  arc  E  E'  actually  described 
in  the  same  time  by  the  earth.  Whence  we  conclude,  that  the 
apparent  motion  of  the  sun  in  space,  and  the  actual  motion  of 
the  earth,  are  the  same  in  every  particular. 

182.  It  has  been  discovered  that  the  motion  of  the  moon  in 
its  revolution  around  the  earth,  is  subject  to  the  same  laws  as  the 
motion  of  a  planet  in  its  revolution  around  the  sun.  We  shall 
assume  this  to  be  a  fact,  and  show  that  our  hypothesis  is  verified 
by  the  results  to  which  it  leads. 

183.  That  point  of  the  orbit  of  a  planet,  which  is  nearest  to 
the  sun,  is  called  the  Perihelion,  and  that  point  which  is  most 
distant  from  the  sun,  the  Aphelion.  The  corresponding  points 
of  the  moon's  orbit,  or  of  the  sun's  apparent  orbit,  are  called,  res- 
pectively, the  Perigee  and  the  Apogee. 

These  points  are  also  called  Apsides  ;  the  former  being  termed 
the  Lower  Apsis,  and  the  latter  the  Higher  Apsis.  The  line 
joining  them  is  denominated  the  Line  of  Apsides. 

The  orbits  of  the  sun,  moon,  and  planets,  being  regarded  as 
ellipses,  the  perigee  and  apogee,  or  the  perihelion  and  aphelion, 
are  the  extremities  of  the  major  axis  of  the  orbit. 

184.  The  law  of  the  angular  motion  of  a  planet  about  the 
sun  may  be  deduced  from  Kepler's  first  law.  Let  V  p  A.p" 
(Fig.  36)  represent  the  orbit  of  a  planet,  considered  as  an 
ellipse,  and  p,  p'  two  positions  of  the  planet  at  two  instants  se- 
parated by  a  short  interval  of  time ;  and  let  71  be  the  middle  point 
of  the  arc  JO  p'.  With  the  radius  S  71  describe  the  small  circular 
arc  1 71  I',  and  with  the  radius  S  b  equal  to  unity,  describe  the  arc 
a  b.  It  is  plain  that  the  two  positions  p,  p'  may  be  taken  so 
near  to  each  other,  that  the  area  S  p  p'  will  be  sensibly  equal  to 
the  circular  sector  S  II'.  If  we  suppose  this  to  be  the  case,  as 
the  measure  of  the  sector  is  ^  I71  I'  x  S  7t  =^  a  b  x  S  71  (substi- 
tuting for  l7t  r  its  value  a  6  x  S  w,)  we  shall  have 

2 

area  Spp'  —  ^abxS7i. 

When  the  planet  is  at  any  other  part  of  its  orbit,  as  n',  if  S  p" 
p'"  be  an  area  described  in  the  same  time  as  before,  we  shall  have 

area  S  p"  p'"  =  ^  a'  6'  X  Sn/". 


80  ASTRONOMY. 

But  these  areas  are  equal  according  to  Kepler's  first  law: 

2  2 

hence,  ^  ah  xS  n  =  ^  a'  b'  xS  n'  .  .  .  .  (37) ; 

2         2 

and  a  b  :  a'  b'  :  :  S  7i' :  S  71, 

that  is,  the  angidar  motion  of  a  planet  about  the  sun  for  a 
short  interval  of  time,  is  inversely/  proportional  to  the  square  of 
the  radhis  vector. 

It  results  from  this  that  the  angular  motion  is  greatest  at  the 
perihelion,  and  least  at  the  aphelion,  and  the  same  at  correspond- 
ing points,  on  either  side  of  the  major  axis :  also,  that  it  de- 
creases progressively  from  the  perihelion  to  the  aphelion,  and 
increases  progressively  from  the  aphelion  to  the  perihelion. 

185.  Now  to  compare  the  true  with  the  mean  angular  motion, 
suppose  a  body  to  revolve  in  a  circle  around  the  sun,  with  the 
mean  angular  motion  of  a  planet,  and  to  set  out  at  the  same 
instant  with  it  from  the  perihelion.  Let  P  M  A  M'  represent  the 
elliptic  orbit  of  the  planet,  and  P  B  a  B'  the  circle  described  by 
the  body.  The  position  B  of  this  fictitious  body  at  any  time 
will  be  the  mean  place  of  the  planet  as  seen  from  the  sun.  The 
two  bodies  will  accomplish  a  semi-revolution  in  the  same  period 
of  time,  and  therefore  be,  respectively,  at  A  and  a  at  the  same 
instant ;  for,  it  is  obvious  that  the  fictitious  body  will  accomplish 
a  semi-revolution  in  half  the  period  of  a  whole  revolution,  and 
by  Kepler's  law  of  areas,  tlie  planet  will  describe  a  semi-ellipse 
in  half  the  time  of  a  revolution.  At  the  outset,  the  motion  of 
the  planet  is  the  most  rapid  (Art.  184),  but  it  continually  de- 
creases until  the  planet  reaches  the  aphelion,  while  the  motion 
of  the  body  remains  constantly  equal  to  the  mean  motion.  The 
planet  will  therefore  take  the  lead,  and  its  angular  distance  ^  S  B 
from  the  body,  will  increase  until  its  motion  becomes  reduced  to 
an  equality  Avith  the  mean  motion,  after  which  it  will  decrease 
until  the  planet  has  reached  the  aphelion  A,  where  it  will  be 
zero.  In  the  motion  from  the  aphelion  to  the  perihelion,  the 
angular  velocity  of  the  planet  will  at  first  be  less  than  that  of 
the  body  (Art.  184),  but  it  will  continually  increase,  while  that 
of  the  body  will  remain  unaltered  :  thus,  the  body  will  now  get 
in  advance  of  the  planet,  and  their  angular  distance  p'  S  B'  will 
increase,  as  before,  until  the  motion  of  the  planet  again  attains 
to  an  equality  with  the  mean  motion,  after  which  it  will  de- 


^ 


DEFINITIONS    OF    TERMS.  81 

crease,  as  before,  until  it  again  becomes  zero  at  the  peri- 
helion. 

It  appears,  then,  that /row  the  'perihelion  to  the  aphelion  the 
true  place  is  in  advance  of  the  mean  place^  and  that  from  the 
aphelion  to  the  perihelion^  on  the  contrary,  the  mean  place  is 
in  advance  of  the  true  place. 

The  angular  distance  of  the  true  place  of  a  planet  from  its 
mean  place,  as  it  would  be  observed  from  the  sun,  is  called  the 
Equation  of  the  Centre.  Thus,  /j  S  B  is  the  equation  of  the 
centre  corresponding  to  the  particular  position  p  of  the  planet. 
It  is  evident,  from  the  foregoing  remarks,  that  the  equation  of 
the  centre  is  zero  at  the  perihelion  and  aphelion,  and  greatest  at 
the  two  points,  as  M  and  M',  where  the  planet  has  its  mean 
motion.  The  greatest  value  of  the  equation  of  the  centre  is 
called  the  Greatest  Equation  of  the  Centre. 

186.  As  the  laws  of  the  motion  of  the  moon  (Art.  182)  and 
of  the  apparent  motion  of  the  sun  (Art.  181)  are  the  same  as 
those  of  a  planet,  the  principles  established  in  the  two  preced- 
ing articles  are  as  applicable  to  these  bodies  in  their  revolution 
around  the  earth,  as  to  a  planet  in  its  revolution  around 
the  sun. 

Definitions  of  Tertns. 

187.  1.  The  Geocentric  Place  of  a  body,  is  its  place  as  seen 
from  the  earth. 

2.  The  Heliocentric  Place  of  a  body,  is  its  place  as  it  would 
be  seen  from  the  sun. 

3.  Geocentric  Longitude  and  Latitude  appertain  to  the  geo- 
centric place,  and  Heliocentric  Longitude  and  Latitude  to  the 
heliocentric  place. 

4.  Two  heavenly  bodies  are  said  to  be  m  Conjunction,  when 
their  longitudes  are  the  same,  and  to  be  in  Opposition.^  when 
their  longitudes  differ  by  180°.  When  any  one  heavenly  body 
is  in  conjunction  with  the  sun,  it  is,  for  the  sake  of  brevity,  said 
to  be  in  Conjunctio7i ;  and  when  it  is  in  opposition  to  the  sun, 
to  be  in  Opposition. 

The  planets  Mercury  and  Venus,  allowing  that  their  distances 

from  the  sun  are  each  less  than  the  earth's  distance  (Art.  19), 

can  never  be  in  opposition.     But  they  may  be  in  conjunction, 

either  by  being  between  the  sun  and  earth,  or  by  being  on  the 

11 


82  ASTRONOMY. 

opposite  side  of  the  sun.  In  the  former  situation  they  are  said 
to  be  in  Inferior  Conjunction,  and  in  the  latter,  in  Superior 
Conjunction. 

5.  A  Synodic  Revolution  of  a  body,  is  the  interval  between 
two  consecutive  conjunctions  or  oppositions. 

For  the  planets  Mercury  and  Venus,  a  synodic  revolution  is 
the  interval  between  two  consecutive  inferior  or  superior  con- 
junctions. 

6.  The  Periodic  Time  of  a  planet,  is  the  period  of  time  in 
which  it  accomplishes  a  revolution  around  the  sun. 

7.  The  Nodes  of  a  planet's  orbit,  or  of  the  moon's  orbit,  are 
the  points  in  which  the  orbit  cuts  the  plane  of  the  ecliptic. 
The  node  at  which  the  planet  passes  from  the  south  to  the 
north  side  of  the  ecliptic  is  called  the  Ascending  Node,  and  is 
designated  by  the  character  Q.  The  other  is  called  the  De- 
scending Node,  and  is  marked  g . 

8.  The  Eccentricity  of  an  elliptic  orbit,  is  the  distance  between 
the  centre  of  the  orbit  and  either  focus. 

Elements  of  the  Orbit  of  a  Planet. 
188.  To  have  a  complete  knowledge  of  the  motions  of  the 
planets,  so  as  to  be  able  to  calculate  the  place  of  any  one  of 
them  at  any  assumed  time,  it  is  necessary  to  know  for  each 
planet,  in  addition  to  the  laws  of  its  motion  discovered  by  Kep- 
ler, the  position  and  dimensions  of  its  orbit,  its  mean  motion, 
and  its  place  at  a  specified  epoch.  These  necessary  particulars 
of  information  are  subdivided  into  seven  distinct  elements, 
called  the  Ele?nents  of  the  Orbit  of  a  Planet,  which  are  as 
follows  : 

1.  The  longitude  of  the  ascending  node. 

2.  The  inclination  of  the  plane  of  the  orbit  to  the  plane  of 
the  ecliptic,  called  the  inclination  of  the  orbit. 

3.  The  mean  distance  of  the  planet  from  the  sun,  or  the  semi- 
major  axis  of  its  orbit. 

4.  The  eccentricity  of  the  orbit. 

5.  The  heliocentric  longitude  of  the  perihelion. 

6.  The  epoch  of  the  planet  being  at  its  perihelion,  or  instead, 
its  mean  longitude  at  a  given  epoch. 

7.  The  periodic  time  of  the  planet. 

The  first  two   ascertain  the  position  of  the  plane  of  the 


ELEMENTS    OF    THE    SUn's    APPARENT    ORBIT.  83 

planet's  orbit ;  the  third  and  fourth,  the  dimensions  of  the  orbit ; 
the  fifth,  the  position  of  the  orbit  in  its  plane  ;  the  sixth,  the 
place  of  the  planet  at  a  given  epoch  ;  and  the  seventh,  its  mean 
rate  of  motion. 

189.  The  elements  of  the  earth's  orbit,  or  of  the  sun's  apparent 
orbit,  are  hut  five  in  number ;  the  first  two  of  the  above  mentioned 
elements  being  wanting,  as  the  plane  of  the  orbit  is  coincident 
with  the  plane  of  the  ecliptic. 

190.  The  elements  of  the  moon's  orbit  are  the  same  with  those 
of  a  planet's  orbit,  it  being  understood  that  the  perigee  of  the 
moon's  orbit  answers  to  the  perihelion  of  a  planet's  orbit,  and  that 
the  geocentric  longitude  of  the  perigee  and  the  geocentric  longi- 
tude of  the  node  of  the  moon's  orbit  answer  respectively  to  the 
heliocentric  longitude  of  the  perihelion,  and  the  heliocentric  lon- 
gitude of  the  node  of  a  planet's  orbit. 

191.  The  linear  unit  adopted,  in  terms  of  which  the  semi-major 
axes,  eccentricities,  and  radii  vectores  of  the  planetary  orbits,  are  ex- 
pressed, is  the  mean  distance  of  the  sun  from  the  earth,  or  the  semi- 
major  axis  of  the  earth's  orbit.  When  thus  expressed,  these  fines 
are  readily  obtained  in  known  measures  whenever  the  mean  dis- 
tance of  the  sun  becomes  known.  The  lines  of  the  moon's  orbit 
are  found  in  terms  of  the  moon's  mean  distance  from  the  earth, 
as  unity. 

Methods  of  Determining  the  Elements  of  the  Sun's  Apparent 

Or'bit,  or  of  the  EartKs  Real  Orbit. 

Mean  motion. 

192.  The  sun's  mean  daily  motion  in  longitude  results  from 
the  length  of  the  mean  tropical  year  obtained  from  observation 
(Art.  176). 

Semi-mcijor  axis. 

193.  As  we  have  just  stated,  the  semi-major  axis  of  the  sun's 
apparent  orbit  is  the  linear  unit,  in  terms  of  which  the  dimensions 
of  the  planetary  orbits  are  expressed.  Its  absolute  length  is  com- 
puted from  the  mean  horizontal  parallax  of  the  sun. 

194.  The  horizontal  parallax  of  a  body  being  given,  to  find 
its  distance  from,  the  earth.     We  have  (equation  9,  p.  43), 

D=-A_; 

Sin  H  ' 
where  H  represents  the  horizontal  parallax  of  the  body,  D  its  dis- 


84  ASTRONOMY. 

tance  from  the  centre  of  the  earth,  and  R  the  radius  of  the  earth. 
The  parallax  of  all  the  heavenly  bodies,  with  the  exception  of  th* 
moon,  is  so  small,  that  it  may,  without  material  error,  be  taken  in 
this  equation  in  place  of  its  sine.     Thus, 

D=-J-j  =  Rx-l    .  .  .  (38). 
sni  H  H 

Ao-ain,  since  6.2831853  is  the  length  of  the  circumference  of  a 

circle,  of  which  the  radius  is  1,  and  1296000  is  the  number  of 

seconds  in  the  circumference,  we  have  6.2831853  :  I  : :  1296000"  : 

X  =  206264".8  =  the  length  of  the  radius  (1)  expressed  in  seconds. 

Hence,  if  the  value  of  H  be  expressed  in  seconds, 

D  =  R  ?!^2648  .  .  .  (39). 
H. 

195.  In  the  determination  of  the  sun's  parallax,  by  the  process 
of  Arts.  100  and  101,  an  error  of  2"  or  3",  equal  to  about  one 
fourth  of  the  whole  parallax,  may  be  committed,  so  that  the  dis- 
tance of  the  sun,  as  deduced  by  equation  (39)  from  his  parallax 
found  in  that  manner,  may  be  in  error  by  an  amount  equal  to  one 
fourth  or  more  of  the  true  distance.  There  is  a  much  more  ac- 
curate method  of  obtaining  the  sun's  parallax,  which  will  be  no- 
ticed hereafter.  It  has  been  found  by  the  method  to  which  we 
allude,  that  the  horizontal  parallax  of  the  sun  at  the  mean  dis- 
tance is  8".58,  which  may  be  relied  upon  as  exact  to  within  a 
small  fraction  of  a  second.  We  have,  then,  for  the  sun's  mean 
distance,  or  the  mean  semi-major  axis  of  his  orbit, 

D  =R?5?^.^-i?  =  24040.19  R  =  95,102,992  miles; 

8".58  '       '  ' 

taldnof  for  R  the  mean  radius  of  the  earth  =  3956  miles. 

o 

JEcce7it7'iciti/. 

196.  First  Method.  By  the  greatest  and  least  daily  motions 
in  longitude.  We  have  already  explained  (Art.  179)  the  mode  of 
deriving  from  observation  the  sun's  motion  in  longitude  from  day 
to  day.  Now,  let  v  =  the  greatest  daily  motion  in  longitude  ;  v'  = 
the  least  daily  motion  in  longitude ;  r  =  the  least  or  perigean  dis- 
tance of  the  sun ;  and  r'  the  greatest  or  apogean  distance  j  and  we 
shall  have,  by  the  principle  of  Art.  174,    ji'-f 

r  -.r' ::  V  v'  :   n/  -y  ; 
whence,        r'  -\-r  \r'  —  r\\  '^  v-\-  V  v'  :  \^v  —  "^  v' : 


ECCENTRICITY    OF    THE    SUn's    APPARENT    ORBIT.  85 


or,  ~     :  r  —  r : :         ^ :   v  v  —  v  -y 

2  2 


semi-major  axis  =  1 ;  and  r'  —  r  =  2  (eccentricity)  -2e. 


but, 

~2~ 

2 

and  ^  =  v^_^      ^  ^  (40) 

The  greatest  and  least  daily  motions  are,  respectively,  (at  a 
mean)  61'.165  and  57'.192.     Substituting,  we  have, 
6  =  0.016791. 

The  eccentricity  may  also  be  obtained  from  the  greatest  and 
least  apjiarent  diameters  by  a  process  similar  to  the  foregoing, 
on  the  principle  that  the  distances  of  the  sun  at  different  times  are 
inversely  proportional  to  his  corresponding  apparent  diameters. 

The  greatest  apparent  diameter  of  the  sun  is  32'  35".6,  and  the 
least  apparent  diameter  31'  31".0. 

197.  Second  Method.     By  the  greatest  equation  of  the  centre. 

1.  To  find  the  greatest  eqicatiori  of  the  centre.  Let  L  =  the 
true  longitude,  and  M  =  the  mean  longitude,  at  the  time  the  true 
and  mean  motions  are  equal  between  the  perigee  and  apogee  ;  L' 
=  the  true  longitude  and  M'  =  the  mean  longitude,  when  the  mo- 
tions are  equal  between  the  apogee  and  perigee ;  and  E  =  the 
greatest  equation  of  the  centre.     Then  (Art.  185), 

L  =  M  +  E,  and  L'  =  M'  — E  ; 
whence,  L'  — L  =M'  — M  — 2  E  ; 

and  E  =  (M'-M)-(L'-L)  ^^^^^ 

About  the  time  of  the  greatest  equation,  the  sun's  true  motion 
and  consequently  the  equation  of  the  centre,  continues  very  neai 
ly  the  same  for  two  or  three  days  ;  we  may,  therefore,  with  bur 
slight  error,  take  the  noon,  when  the  sun  is  on  either  side  of  the 
line  of  apsides,  that  separates  the  two  days  on  which  the  motions 
in  longitude  are  most  nearly  equal  to  59'  8",  as  the  epoch  of  the 
greatest  equation. 

The  longitude  L  or  L'  at  either  epoch  thus  ascertained,  results 


86  ASTRONOMY. 

from  the  observed  right  ascension  and  declination.  M'  —  M  =  the 
mean  motion  in  longitude  in  the  interval  of  the  epochs,  and  is 
found  by  multiplying  the  number  of  mean  solar  days  and  fractions 
of  a  day  comprised  in  the  interval,  by  59'  8".330,  the  mean  daily 
motion  in  longitude. 

For  example.  From  observations  upon  the  sun,  made  by  Dr. 
Maskelyne  in  the  year  1775,  it  is  ascertained  in  the  manner  just 
explained,  that  the  sun  was  near  its  greatest  equation  at  noon,  or 
at  Oh.  3m.  35s.  mean  solar  time  on  the  2d  April,  and  at  noon  on 
the  31st,  or  at  23  h.  49  m.  35  s.  mean  solar  time  on  the  30th  of 
September.  The  observed  longitudes  were,  at  the  first  period, 
12°  33'  39".06,  and  at  the  second  188°  5'  44".45.  The  interval  of 
time  between  the  two  epochs  is  182  d.  —  14  m. 

Meanmotioninl82d.  — 14m.  .     .     179°  22'  41".56 
Difference  of  two  longitudes  .     .     .     175    32      5.39 


Diff. 2  )   3    50   36  .17 


Greatest  equation  of  centre     ...         1    55    18  .08 

More  accurate  results  are  obtained,  by  reducing  observations 
made  during  several  days  before  and  after  the  epoch  of  the  great- 
est equation,  and  taking  the  mean  of  the  different  values  of  the 
greatest  equation  thus  obtained.  According  to  M.  Delambre,  the 
greatest  equation  was  in  1775,  1°  55'  31".66. 

2.   Tlie  eccentricity  of  an  orbit  may  be  derived  from  the  great- 
est equation  of  the  centre  by  means  of  the  following  formula  : 
^_K_11K^_587K^_ 
-  2         "3:2^     "3:5:2^       '^'-  •  •  ^^'^^' 

in  which  K  stands  for  the  expression f  E  beino-  the 

^  57°.2957795   ^  =' 

greatest  equation  of  the  centre).     In  the  case  of  the  sun's  orbit,  K 
being  a  small  fraction,  all  its  powers  beyond  the  first  may  be  omit- 
ted.    Thus,  retaining  only  the  first  term  of  the  series,  and  taking 
E  =  1°  55'  31".66  the  greatest  equation  in  1775,  we  have, 
K         1°  55'  31".66  ...„„., 

'  -  2  =2^^57^295779-5-  =  •^^^^'^^- 
198.  From  observations  made  at  distant  periods,  it  is  discovered 
that  the  equation  of  the  centre,  and  consequently  the  eccentricity, 
is  subject  to  a  continual  slow  diminution.     The  amount  of  the 


PERIGEE    OF    THE    SUN's    APPARENT    ORBIT.  87 

diminution  of  the  greatest  equation  in  a  centuiy,  called  the  secular 
diminution,  is  estimated  by  Delambre,  at  17".2. 

Longitude  and  epoch  of  the  jjerigee. 

199.  As  the  sun's  angular  velocity  is  the  greatest  at  the  perigee, 
the  longitude  of  the  sun  at  the  time  its  angular  velocity  is  greatest, 
will  be  the  longitude  of  the  perigee.  The  time  of  the  greatest 
angTilar  velocity  may  easily  be  obtained  within  a  few  hours,  by 
means  of  the  daily  motions  in  longitude,  derived  from  observation 
(Art.  178). 

200.  The  more  accurate  method  of  determining  the  longitude 
and  epoch  of  the  perigee,  rests  upon  the  principle  that  the  apogee 
and  perigee  are  the  only  two  points  of  the  orbit  whose  longitudes 
differ  by  180°,  in  passing  from  one  to  the  other  of  which  the  sun 
employs  just  half  a  year.  This  principle  may  be  inferred  from 
Kepler's  law  of  areas,  for  it  is  a  well  known  property  of  the  ellipse, 
that  the  major  axis  is  the  only  line  drawn  through  the  focus,  that 
divides  the  ellipse  into  equal  parts,  and  by  the  law  in  question 
equal  areas  correspond  to  equal  times. 

201.  By  a  comparison  of  the  results  of  observations  made  at 
distant  epochs,  it  is  discovered  that  the  longitude  of  the  perigee 
is  continually  increasing  at  a  mean  rate  of  61". 5  per  year.  As 
the  equinox  retrogrades  50".2  in  a  year,  the  perigee  must  then 
have  a  direct  motion  in  space  of  11".3  per  year. 

It  will  be  seen,  therefore,  that  the  interval  between  the  times 
of  the  sun's  passage  through  the  apogee  and  perigee  is  not,  strictly 
speaking,  half  a  sidereal  year,  but  exceeds  this  period  by  the 
interval  of  time  employed  by  the  sun  in  moving  through  an  arc 
of  5".6  the  motion  of  the  apogee  and  perigee  in  longitude  in  half 
a  year. 

202.  According  to  the  most  exact  determinations,  the  mean 
longitude  of  the  perigee  of  the  sun's  orbit  at  the  beginning  of  the 
year  1800,  was  279°  30'  8".39. 

203.  The  heliocentric  longitude  of  the  perihelion  of  the  earth's 
orbit,  is  equal  to  the  geocentric  longitude  of  the  perigee  of  the 
sun's  apparent  orbit  minus  180°.  For,  let  A  E  P  (Fig.  35)  be  the 
earth's  orbit,  and  P  V  the  direction  of  the  vernal  equinox.  Wlien 
the  earth  is  in  its  perihelion  P,  the  sun  is  in  its  perigee  S,  and  we 
have  the  heliocentric  longitude  of  the  perihelion,  VSP=VPL  = 


00  ASTRONOMY. 

angle  ab  c  — 180^  =  geocentric  longitude  of  the  sun's  perigee  — 
180°.* 

204.  The  epoch  and  mean  longitude  of  the  perigee  of  the  sun's 
orbit  being  once  found,  the  sun's  mean  longitude  at  any  assumed 
epoch  is  easily  obtained  by  means  of  the  mean  motion  in  lon- 
gitude. 

Methods  of  determining  the  Elements  of  the  Mooris  Orbit. 
Longitude  of  the  node. 

205.  In  order  to  obtain  the  longitude  of  the  moon's  ascending 
node,  we  have  only  to  find  the  longitude  of  the  moon  at  the 
time  its  latitude  is  zero  and  the  moon  is  passing  from  the  south 
to  the  north  side  of  the  ecUptic  ;  and  this  may  be  deduced 
from  the  longitudes  and  latitudes  of  the  moon,  derived  from 
observed  right  ascensions  and  declinations,  by  methods  precisely 
analogous  to  those  by  which  the  right  ascension  of  the  sun, 
at  the  time  its  declination  is  zero  and  it  is  passing  from  the  south 
to  the  north  of  the  equator,  or  the  position  of  the  vernal  equinox, 
is  ascertained.     (See  Art.  169.) 

Inclination  of  the  orbit. 

206.  Amongst  the  latitudes  computed  from  the  moon's  observ^ed 
right  ascensions  and  declinations,  the  greatest  measures  the  incli- 
nation of  the  orbit.  It  is  found  to  be  about  5° ;  sometimes  a 
little  greater,  and  at  other  times  a  little  less. 

Mean  onotion. 
■  207.  With  the  longitudes  of  the  moon,  found  from  day  to  day,  it  is 
easy  to  obtain  the  interval  from  tlie  time  at  which  the  moon  has 
any  given  longitude  till  it  returns  to  the  same  longitude  again. 
This  interval  is  called  a  Tropical  Revolution  of  the  moon.  It  is 
found  to  be  subject  to  considerable  periodical  variations,  and  thus 
one  observed  tropical  revolution  may  differ  materially  from  the 
mean  period.  In  order  to  obtain  the  mean  tropical  revolution,  we 
must  compare  two  longitudes  found  at  distant  epochs.  Their 
diiference,  augmented  by  the  product  of  360°  by  the  number  of 
revolutions  performed  in  the  interval  of  the  epochs,  will  be  the 
mean  motion  in  longitude  in  the  interval,  from  which  the  mean 


*  It  is  plain  that  the  same  relation  subsists  between  the  heliocentric  longitude 
of  the  oartli  and  the  geocentric  longitude  of  the  sun  in  every  other  position  of  the 
earth  in  its  orbit. 


moon's  mean  motion  in  longitude.  89 

motion  in  100  years  or  36525  days,  called  the  Secular  motion, 
may  be  obtained  by  a  simple  proportion.  The  secular  motion 
being  once  known,  it  is  easy  to  deduce  from  it  the  period  in  which 
the  motion  is  360°,  which  is  the  mean  topical  revolution. 

It  should  be  observed,  however,  that  to  find  the  precise  mean 
secular  motion  in  longitude,  it  is  necessary  to  compare  the  mean 
longitudes  instead  of  the  true.  Now,  the  true  longitude  of  the 
moon  at  any  time  having  been  found,  the  mean  longitude  at  the 
same  time  is  derived  from  it,  by  correcting  for  the  equation  of  the 
centre,  and  certain  other  periodical  inequalities  of  longitude  here- 
after to  be  noticed.  But  this  cannot  be  done,  even  approximately, 
until  the  theory  of  the  moon's  motions  is  known  with  more  or  less 
accuracy. 

208.  The  longitude  of  the  moon,  at  certain  epochs,  may  be  very 
conveniently  deduced  from  observ^ations  upon  lunar  eclipses. 
For,  the  time  of  the  middle  of  the  eclipse  is  very  near  the  time  of 
opposition  when  the  longitude  of  the  moon  differs  180°  from  that 
of  the  sun,  and  the  longitude  of  the  sun  results  from  the  known 
theory  of  its  motions.  The  recorded  observations  of  the  ancients 
upon  the  times  of  the  occurrence  of  eclipses,  are  the  only  obser- 
vations that  can  now  be  made  use  of,  for  the  direct  determination 
of  the  longitude  of  the  moon  at  an  ancient  epoch. 

209.  The  mean  tropical  revolution  of  the  moon  is  found  to  be 

27.321582d.  or  27d.  7h.  43m.  4.7s. 
Hence  27.321.582 d.  :  Id.  :  :  360°  :  13°.17639.  =  13°  10'  35".0  = 
moon's  mean  daily  motion  in  longitude. 

210.  Since  the  equinox  has  a  retrograde  motion,  the  sidereal 
revolution  of  the  moon  must  exceed  the  tropical  revolution,  as  the 
sidereal  year  exceeds  the  tropical  year.  The  excess  will  be  equal 
to  the  time  employed  by  the  moon,  in  describing  the  arc  of  pre- 
cession answering  to  a  revolution  of  the  moon.     Thus, 

365.25d. :  50".2  : :  27.3d. :  3".75  =  arc  of  precession, 
and  13°.17  : :  Id. : :  3  ".75  :  6.9s.  =  excess. 

Wherefore,  the  mean  sidereal  revolution  of  the  moon  is  27 d.  7h>, 
43m.  11.6s. 

211.  It  has  been  found,  by  determining  the  moon's  mean  rate  of 
motion  for  periods  of  various  lengths,  that  it  is  subject  to  a  con- 
tinual slow  acceleration.  This  acceleration  will  not,  however,  be 
indefinitely  progressive :    Laplace  has  investigated  its  physical 

12 


90  ASTRONOMY. 

cause,  and  shown  from  the  principles  of  Physical  Astronomy,  that 
it  is  really  a  periodical  inequality  in  the  moon's  mean  motion, 
which  requires  an  immense  length  of  time  to  go  through  its  dif- 
ferent values. 

The  mean  motion  given  in  Art.  209  answers  to  the  commence- 
ment of  the  present  century. 
Longitude  of  the  perigee^  eccentricity^  and  semi-major  axis. 

212.  The  methods  of  determining  these  elements  of  the  moon's 
orbit,  are  similar  to  those  by  which  the  corresponding  elements 
of  the  sun's  orbit  are  found.  It  is  to  be  observed,  however,  that 
for  the  longitudes  of  the  sun,  which  are  laid  off  in  the  plane 
of  the  ecliptic,  in  the  case  of  the  moon,  corresponding  angles 
are  laid  off  in  the  plane  of  its  orbit.  These  angles  are  reckoned 
from  a  line  drawn  making  an  angle  with  the  line  of  nodes 
equal  to  the  longitude  of  the  ascending  node,  and  are  called 
Orbit  Longitudes.  The  orbit  longitude  is  equal  to  the  moon's 
angular  distance  from  the  ascending  node  plus  the  longitude  of 
the  ascending  node. 

213.  The  ecliptic  longitude  of  the  moon  at  any  tim,e  being 
given,  to  find  the  orbit  longitude. 

Let  V  N  M  (Fig.  38)  represent  the  moon's  orbit  referred  to 

the  sphere  of  the  heavens,  V  N  m  the  ecliptic,  V  the  vernal 

equinox,  and  V  the  corresponding  point  from  which  the  orbit 

longitudes  are  reckoned.     The  orbit  longitude  V  N  M  =  V  N  + 

NM  =  VN  +  NM  =  long,  of  node  +  N  M.     Now,  by  Napier's 

rules, 

cos  M  N  m.  =  cot  N  M  tang  N  m ; 

or,  cot  N  M  -  cos  M  N  m,  cot  N  m  .  .  .  (42). 

N  m  =  ecliptic  long.  —  long,  of  node  ;  and  M  N  m  =  inclina- 
tion of  orbit. 

214.  The  horizontal  parallax  of  the  moon,  like  almost  every 
other  element  of  astronomical  science,  is  subject  to  periodical 
changes  of  value.  It  varies  not  only  during  one  revolution,  but 
also  from  one  revolution  to  another.  The  fixed  and  mean 
parallax  about  which  the  true  parallax  may  be  conceived  to 
oscillate,  answers  to  the  mean  distance,  that  is,  the  distance 
about  which  the  true  distance  varies  periodically,  and  is  called 
the  Constant  of  the  Parallax.  It  is,  for  the  equatorial  radius 
of  the  earth,  57'  0".9  ;  from  which  we  find  by  equation  (39)  the 


DETERMINA'J  ION    OF    THE    ELEMENTS.  91 

mean  distance  of  the  moon  from  the  earth  to  be  60.3  radii,  or 
about  240,000  miles. 

The  greatest  and  least  parallaxes  of  the  moon  are  61'  24"  and 
53'  48". 

215.  The  eccentricity  of  the  moon's  orbit  is  more  than  three 
times  as  great  as  that  of  the  sun's  orbit.  Its  greatest  equation 
exceeds  6°. 

Mean  longitude  at  an  assigned  epoch. 

216.  We  have  already  explained  (Art.  207)  the  principle  of 
the  determination  of  the  mean  longitude  of  the  moon  from  an 
observed  true  longitude.  Now,  when  the  mean  longitude  at 
any  one  epoch  whatever  becomes  known,  the  mean  longitude  at 
any  assigned  epoch  is  easily  deduced  from  it  by  means  of  the 
mean  motion  in  longitude. 

Methods  of  determining  the  Elemetits  of  a  Planefs  Orbit. 

217.  The  methods  of  determining  the  elements  of  the  plane- 
etary  orbits  suppose  the  possibility  of  finding  the  heliocentric 
longitude  and  the  radius  vector  of  the  earth  for  any  given  time. 
Now,  the  elements  of  the  earth's  orbit  having  been  found  by  the 
processes  heretofore  detailed,  the  longitude  may  be  computed  by 
means  of  Kepler's  first  law,  and  the  radius  vector  from  the 
polar  equation  of  the  elliptic  orbit.  The  manner  of  effect- 
ing such  computation  will  be  considered  hereafter,  at  present 
the  possibility  of  effecting  it  will  be  taken  for  granted. 

Heliocentric  longitude  of  the  ascending  node. 

218.  When  the  planet  is  in  either  of  its  nodes,  its  latitude  is- 
zero.  It  follows,  therefore,  that  the  longitude  of  the  planet  at 
the  time  its  latitude  is  zero,  is  the  geocentric  longitude  of  the 
node  at  the  time  the  planet  is  passing  through  it.  Now,  if  the 
right  ascension  and  declination  of  the  planet  be  observed  from 
day  to  day,  about  the  time  it  is  passing  from  one  side  of  the 
ecliptic  to  the  other,  and  converted  into  longitude  and  latitude, 
the  time  at  Avhich  the  latitude  is  zero,  and  the  longitude  at  that 
time  may  be  obtained  by  a  proportion.  When  the  planet  is 
again  in  the  same  node,  the  geocentric  longitude  of  ihe  node 
may  again  be  found  in  the  same  manner  as  before.  On  account 
of  the  different  position  of  the  earth  in  its  orbit,  this  longitude 
will  differ  from  the  former. 


92  ASTRONOMY. 

Now,  if  two  geocentric  longitudes  of  the  same  node  he  founds 
its  heliocentric  longitude  may  be  computed. 

Let  S  (Fig.  39)  be  the  sun,  N  the  node,  and  E  one  of  the 
positions  of  the  earth  for  which  the  geocentric  longitude  of  the 
node  V  E  N  is  known.  Denote  this  angle  by  G,  the  sun's  lon- 
gitude V  E  S  by  S,  and  the  radius  vector  S  E  by  r.  Also,  let  E' 
be  the  other  position  of  the  earth,  and  denote  the  corresponding 
quantities  for  this  position,  V  E'  N,  V  E'  S,  and  S  E',  respectively, 
by  G',  S',  and  r'.  Let  the  radius  vector  of  the  planet  when  in 
its  node,  or  S  N  =  V  ;  and  the  heliocentric  longitude  of  the  node, 
or  V  S  N  =  X.     The  triangle  S  N  E  gives, 

sin  S  N  E  :  sin  S  E  N  :  :  S  E  :  S  N ; 
but  S  E  N  =  V  E  S  —  V  E  N  =  S  —  G  ; 

and,  SNE  =  VAN— VSN=VEN  —  VSN  =  G  —  X; 
hence,  sin  (G  —  X)  :  sin  (S  —  G)  : :  r  :  Y  ; 

or,  rsin(S  — G)=Vsin(G  — X)  .  .  .  (43). 

In  like  manner,  r'  sin  (S'  —  G')  -  V  sin  (G'  —  X) ; 


dividing. 


r  sin  (S  —  G)  _  sin  (G  —  X) 


or, 


r'sin(S'  — G')      sin  (G' —  X) ' 
T  sin  (S  —  G)  _  sin  G  cos  X  —  sin  X  cos  G 


r'  sin  (S'  —  G')      sin  G'  cos  X  —  sin  X  cos  G' 

sin  G  —  cos  G  tang  X  . 
sin  G'  —  cos  G'  tang  X  ' 
whence, 

^         r  sin  (S  —  G)  cos  G'  —  r'  sin  (S'  —  G')  cos  G  "  '  ^     '' 

Equation  (43)  gives  ¥  =  "1^^^=^-)  .  .  .  (45). 

sni  (G  —  X) 

Inclination  of  the  orbit. 
219.  The  longitude  of  the  node  having  been  found  by  the 
preceding  or  some  other  method,  compute  the  day  on  which  the 
sun's  longitude  will  be  the  same  or  nearly  the  same :  the  earth 
will  then  be  on  the  line  of  the  nodes.  Observe  on  that  day  the 
planet's  right  ascension  and  declination,  and  deduce  the  geo- 
centric longitude  and  latitude.  Let  E  N  />  (Fig.  40)  be  the  plane 
of  the  ecliptic,  V  the  vernal  equinox,  S  the  sun,  N  the  node,  E 
the  earth  on  the  line  of  nodes,  and  P  the  planet  as  referred  to 


PERIODIC    TIME, REDUCTION    OP    OBSERVATIONS.  93 

the  celestial  sphere,  from  the  earth.  Let  X  denote  the  geocentric 
latitude  Pj9  ;  E  the  arc  N  p  =  V  /9  — V  N  =  geo.  long,  of  planet  — 
long,  of  node  ;  and  I  the  inclination  P  Np.  The  right  angled 
triangle  P  N  /j  gives 

sin  N  />  =  tang  P  p  cot  P  N  2>  =  tang  X  cot  I ; 

hence,      cot  I  = ,  and  tang  I  =  —^^    .  .  •  (46). 

'  tangx'  ^         sinE  ^     ^ 

220.  It  will  be  understood,  that  to  obtain  an  exact  result,  we 
must  compute  the  precise  time  of  the  day  at  which  the  longitude 
of  the  sun  is  the  same  as  that  of  the  node,  and  then,  by  means  of 
their  observed  daily  variations,  correct  the  longitude  and  latitude 
of  the  planet  for  the  variations  in  the  interval  between  the  time 
thus  ascertained  and  the  time  of  the  observation  above  men- 
tioned. 

Periodic  time. 

221.  The  interval  from  the  time  the  planet  is  in  one  of  its 
nodes  till  its  return  to  the  same,  gives  the  periodic  time  or  side- 
real revolution. 

222.  Another  and  more  accurate  method  is  to  observe  the 
length  of  a  synodic  revolution  (p.  82),  and  to  compute  the  pe- 
riodic time  from  this.  If  we  compare  the  time  of  a  conjunction 
which  has  been  observed  in  modern  times,  with  that  of  a  con- 
junction observed  by  the  earlier  astronomers,  and  divide  the  in- 
terval between  them  by  the  number  of  synodic  revolutions  con- 
tained in  it,  we  shall  have  the  mean  synodic  revolution  with 
great  exactness,  from  which  the  mean  periodic  time  may  be  de- 
duced.* 

To  find  the  heliocentric  longitude  and  latitude,  and  the  radius 
vector,  for  a  given  time. 

223.  The  earth  being  in  constant  motion  in  its  orbit,  and  be- 
ing thus  at  different  times  very  differently  situated  with  regard 
to  the  other  planets,  as  well  in  respect  to  distance  as  direction, 
it  is  necessary  for  the  purpose  of  comparing  the  observations 
made  upon  these  bodies  with  each  other,  to  refer  them  all  to  one 
common  point  of  observation.  As  the  sun  is  the  fixed  centre 
about  which  the  revolutions  of  the  planets  are  performed,  it  is 


*  We  shall,  in  the  sequel,  investigate  the  equation  that  expresses  the  relation 
between  the  synodic  revolution  and  the  periodic  time. 


94  ASTRONOMY. 

the  point  best  suited  to  this  purpose,  and  accordingly  it  is  to  the 
sun  that  the  observations  are  in  reality  referred.  The  reduction 
of  observations  from  the  earth  to  the  sun,  as  it  is  actually  per- 
formed, consists  in  the  deduction  of  the  heliocentric  longi- 
tude and  latitude  from  the  geocentric  longitude  and  latitude, 
these  beina:  derived  from  the  observed  riffht  ascension  and  de- 
clination.  We  will  now  show  how  to  effect  this  deduction, 
supposing  that  the  longitude  of  the  node  and  the  inclination  of 
the  orbit  are  known.  Let  N  P  (Fig.  41)  be  part  of  the  orbit  of  a 
planet,  S  N  C  the  plane  of  the  ecliptic,  N  the  ascending  node,  S 
the  sun,  E  the  earth,  and  P  the  planet ;  also,  let  P  *  be  a  per- 
pendicular let  fall  from  P  upon  the  plane  of  the  ecliptic,  and  E 
V,  S  V  the  direction  of  the  vernal  equinox.  Let  X  :=  P  E  *  the 
geocentric  latitude  of  the  planet ;  /  =  P  S  *  its  heliocentric  lati- 
tude ;  G  =  V  E  "TT  its  geocentric  longitude  ;  L  =  V  S  tt  its  helio- 
centric longitude  ;  S  =  V  E  S  the  longitude  of  the  sun  ;  N  =^  V  S 
N  the  heliocentric  longitude  of  the  node  ;  I  =  P  N  C  the  inclina- 
tion of  the  orbit ;  r  =  S  E  the  radius  vector  of  the  earth  ;  and  v 
=  S  P  the  radius  vector  of  the  planet. 

The  point  -r  is  called  the  reduced  place  of  the  planet,  and  S  -r 
its  curtate  distance.  All  the  angles  of  the  triangle  S  E  *  have 
also  received  particular  appellations  :  S  *  E  the  angle  subtended 
at  the  reduced  place  of  the  planet  by  the  radius  of  the  earth's  orbit, 
is  called  the  Annual  Parallax^  S  E  *  the  Elongation,  and  E  S 
ir  the  Commutation.  Let  A=S'rE,  E=SE'7r,  and  C  =  E  S  *. 
Draw  S  -tt'  parallel  to  E  * :  then  A  =  *S';r'  =  VS*— VS^'  = 
VS*— VE':r  =  L  — G;  E  =  V  E  *  — VE  S  =  G  —  S  ;  C  =  V 
S  E  —  V  S  *  =  180°  +  V  S  E'  —  V  S  *  =  180°  +  V  E  S  —  V  S 
*  =  180°  +  S  —  L  =  T  —  L  (putting  T  =  180°  +  S). 

1.  For  the  latitude.     The  triangles  E  P  •tt,  S  P  *  give 

E  -r  tang  X  =  P  *  =  S  ^  tang  I,  whence  ^'^"^  ^  =  ?-l  • 

tang  I       E  * 

but,  S*:E*:  :  sin  E  :  sin  C,  or,  S^  ^sin_E 

'     '  E*      sin  C 

substituting,  toig_X^sinE 

tang  I      sin  C 

whence,  tang  X  sin  C  =  tang  Z  sin  E  .  .  .  (46). 

or,         tang  X  sin  (T  —  L)  =  tang  I  sin  (G  —  S)  .  .  .(47). 


REDUCTION    OF    OBSERVATIONS    TO    THE    SUN.  95 

Again,  the  triang-le  N  P  p  gives,  by  Napier's  rules, 

sin  N  p  =  cot  P  N  P  tan  P  p,  or,  sin  (L  —  N)  =  cot  I  tan  /  .  (48). 

Either  of  the  equations  (47)  and  (48)  will  give  the  value  of  /, 

when  the  longitude  L  is  known. 

2.  For  the  longitude.     If  we  substitute  in  equation  (47)  the 

value  of  tang  Z,  given  by  equation  (48),  and  replace  (G  —  S)  by 

E,  we  have, 

tang  X  sin  (T  —  L)  =  sin  (L  —  N)  tang  I  sin  E  ; 

but  T  —  L  =  (T  —  N)  —  (L  —  N)  =  D  —  (L  —  N),  (denoting  (T 

—  N)  by  D)  ;  substituting  and  designating  L  —  N  by  :r, 

tang  X  sin  (D  —  x)—  sin  x  tang  I  sin  E  ; 

whence, 

tang  X  sin  D  cos  x  —  tang  X  cos  D  sin  x  =  tang  I  sin  E  sin  x ; 

or,  tang  X  sin  D  —  tang  X  cos  D  tang  x  =  tang  I  sin  E  tang  x  ; 

1,-  u     •  4-  tang  X  sin  D  z^nx 

which  gives,     tang.r  = =- — — : — — -    .  .  .  (49.) 

^  ^        tang  X  cos  D  +  tang  I  sin  E  ^      ^ 

Substituting  the  values  of  x,  D  and  E,  we  have  finally, 

tano-  fL  — N)  = tang  X  sin  (T  —  N) .5.. 

*^  ^      tangXcos(T  — N)+tangIsin(G  — S)    ^     ^' 

As  N  is  known,  the  value  of  L  will  result  from  this  equation. 

224.  The  co-ordinates  employed  to  fix  the  position  of  a  planet 
in  the  plane  of  its  orbit,  are  its  orbit  longitude  (Art.  212)  and  its 
radius  vector,  both  of  which  result  from  the  heliocentric  longitude 
and  latitude,  the  longitude  of  the  node  and  the  inclination  of  the 
orbit  being  known.  In  Fig.  41,  V  N  P  represents  the  orbit  longi- 
tude, and  S  P  (=  ■?;)  the  radius  vector,  for  the  position  P.  Now, 
the  triangle  P  S  -^  gives, 

P  =  =^  —  ,  or,  V  = _  ; 

cos  r  o  *  cos  / 

and  the  triangle  E  S  *  gives, 

•     A      •    T7    o  T-<    o        S  E  sin  E     r  sin  E 

sm  A  :  sin  E  :  S  E  :  S  *  = =  — .. ; 

sm  A  sin  A 

whence,  by  substitution,  v  =  —. — ^ =  -; — — — L_IZ — L(5l). 

sin  A  cos  Z       sm(L — G)cosZ 

The  orbit  longitude  L'  =N  P  +long.  of  node  .  .  .  (52). 

And  to  find  N  P,  the  triangle  N  P  p  gives, 


tangN 
cos  I 
and,  N /)  =  long,  of  planet  —  long,  of  node  .  .  .  (52). 


cos  P  N  p  =  cot  N  P  tang  N  p,  or  tang  N  P  =  ^^^}SIlP  .  .  .  (52) ; 


1)6  ASTllONOMY. 

225.  The  heliocentric  longitude  may  be  obtained  in  a  very 
simple  manner,  if  the  observations  be  made  upon  the  planet  at 
the  time  of  conjunction  or  opposition ;  for,  it  will  then  either 
be  equal  to  the  geocentric  longitude  or  differ  180°  from  it. 

When  the  heliocentric  longitude  is  found,  the  latitude  may 
be  computed  by  equation  (47)  or  (48).  Equation  (51)  will  disap- 
pear in  conjunctions  and  oppositions  ;  but  the  radius  vector 
(S  P)  may  be  computed  from  the  triangle  ESP  (Fig.  42) :  for, 
the  side  S  E  the  radius  vector  of  the  earth,  is  knoAvn,  as  well  as 
the  angle  SEP  the  geocentric  latitude  of  the  planet,  and  the 
angle  E  S  P  =  180°  —  VSp  =  180°  —  heliocentric  kit. 

226.  The  radius  vector  of  either  of  the  inferior  planets*  at  the 
time  of  maximum  elongation,  may  be  approximately  deduced 
from  the  amount  of  the  maximum  elongation,  determined  from 
observation.  The  elongation  of  an  inferior  planet  at  any  time, 
is  equal  to  the  difference  of  the  geocentric  longitudes  of  the  plan- 
et and  sun,  and  is  therefore  easily  obtained.  Let  N  P  P'  (Fig.  43) 
represent  the  orbit  of  an  inferior  planet,  supposed  to  lie  in  the 
plane  of  the  ecliptic.  The  line  E  P  drawn  from  the  earth  to 
the  planet,  will  at  the  time  of  maximum  elongation  be  tangent 
to  the  orbit ;  and  thus,  if  the  greatest  value  of  the  elongation 
be  observed,  we  shall  have  in  the  rig-ht  ans^led  triangle  EPS. 
the  line  E  S,  and  the  angle  SEP,  from  which  the  radius  vector 
S  P  may  be  computed. 

As  the  earth  and  planet  are  in  motion,  the  greatest  elon- 
gation will  occur  at  different  points  of  the  planet's  orbit,  and 
therefore  we  may  find  by  the  foregoing  process  different  radii 
vectores. 

Longitude  of  the  j)erihelionj  eccentricity,   and  semi-major 

axis. 

227.  The  longitude  of  the  perihelion,  the  eccentricity,  and 
the  semi-major  axis,  may  be  derived  from  the  heliocentric  orbit 
longitude  and  the  radius  vector  found  for  three  different  times. 

Let  S  P,  S  P',  S  P"  (Fig.  44)  be  the  three  given  radii  vectores, 
V  S  P,  V  S  P',  V  S  P",  the  three  given  longitudes,  and  A  B 
the  line  of  apsides  of  the  planet's  orbit.  Let  the  ansfles  P  S  P', 
P  S  P",   which   are  known,  be  represented  by  w,  w,  and  the 

*  An  Inferior  planet  is  one  whose  orbit  lies  within  the  orbit  of  the  earth. 


LONGITUDE    OF    THE    PERIHELION,    &C.  97 

angle  B  S  P,  which  is  unknown,  by  x ;  and  let  the  three  radii 
vectores  S  P,  S  P',  S  P"  be  denoted  by  v,  v',  v"  ;  the  semi-major 
axis  A  C  by  a,  and  the  ratio  of  the  eccentricity  to  the  semi-major 
axis  by  e :  then,  the  three  unknown  quantities  which  are  to  be 
determined,  are  a,  e,  and  the  angle  x,  and  the  general  polar 
equation  of  the  ellipse  furnishes  for  their  determination  the 
three  equations : 

v=   ^g-^^)    .  .  .(53). 
1  +  e  cos  X 

v'=        «(1ZZ_^!)__    .  ,  .  (r4). 
1  +  e  cos  {x  +  m) 

v"  =       <»  (1  —  e')         ^  _  _  (55^ 
1  +  e  cos  {.V  +  n) 

Equating  the  values  of  a  (I  —  e^)  obtained  from  equations  (53, 
and  (54),  we  have, 

V  -\-v  e  cos  X  —  v'  -\-v'  e  cos  {x  +  m), 

or,  e  = .  .  .  (56). 

V  cos  X  —  v'  cos  [x  -f-  m)  k-^ 

In  like  manner  from  (53)  and  (55), 

'""""'*'  .  (57). 


V  cos  X  —  v"  COS  {x  -\-  n) 
Let  v'  —  v  =  p,  and  v"  —  v  =  q  ;  then,  by  equating  the  second 
members  of  equations  (56),  (57),  and  transforming,  we  obtain, 
p  _  V  cos  X  —  v'  cos  {x  +  m) 
q      V  cos  X  —  v"  cos  {x  -f-  n) 

_  V  cos  X  —  v'  cos  7n  cos  x  -\-  v'  sin  m  sin  x 
V  cos  X  —  v"  cos  71  cos  X  +  ■!;"  sin  w  sin  x 

_  V  —  v'  cos  ni  -{-  v'  sin  m  tang  a; 
■y  —  v"  cos  w.  -|-  v"  sin  n  tang  ar ' 

whence,  tang  a:  =  ^  (^  "  ^"  ^^.^  ^)  "jl^  -  ^'  "^^  ^^l  .(58). 

q  v'  sin  wi  —  p  V   sm  ?i 

The  value  of  x  being  found  by  this  equation,  and  subtracted 
from  the  orbit  longitude  of  the  planet  in  the  first  position  P, 
the  result  will  be  the  orbit  longitude  of  the  perihelion.  Also, 
X  being  known,  e  may  be  computed  from  either  of  the  equa- 
tions (56)  and  (57).  And  hence  again,  the  semi-major  axis  from, 
equation  (53),  (54),  or  (55). 
13 


98  ASTRONOMY. 

228.  The  semi-major  axis  or  mean  distance  from  the  sun, 
may  also  be  had  by  taking  the  mean  of  a  great  number  of  radii 
vectores  found  for  every  variety  of  position  of  the  planet  in  its 
orbit. 

229.  Now  that  Kepler's  third  law  has  been  established  by 
investigations  in  Physical  Astronomy,  it  furnishes  the  most  ac- 
curate method  of  finding  the  mean  distance  of  a  planet  from  the 
sun.  Thus,  let  P  =  the  periodic  time  of  the  planet,  and  a  =  its 
mean  distance  ;  then,  the  length  of  the  sidereal  year  being 
365.250374  days  (Art.  177), 

(365.256374d.)^:P*::  P:a^; 

/  P  \- 

whence,  a  =  I 1 3  .  ,  .  (59). 

\365.256374d./  ^     ' 

Epoch  of  a  planet  being  at  the  perihelion  of  its  orhit. 

229.  From  several  observations  upon  the  planet,  about  the 
time  it  has  the  same  longitude  as  the  perihelion,  the  correct 
time  of  its  being  at  the  perihelion  may  be  easily  determined  by 
proportion, 

230.  The  mean  longitude  at  an  assigned  epoch  is  obtained 
upon  the  same  principles  as  the  mean  longitude  of  the  sun  or 
moon  (Arts.  204  ,207.) 

Remarks. 

231.  The  foregoing  methods  of  determining  the  elements  of  a 
planet's  orbit  suppose  observations  to  be  made  at  two  or  more 
successive  returns  of  the  planet  to  its  node.  It  is  possible,  by 
certain  methods  of  trial  and  conjecture,  to  derive  the  elements 
of  a  planetary  orbit,  with  tolerable  accuracy,  from  observations 
continued  during  only  a  part  of  a  revolution,  and  without  wait- 
ing for  a  passage  of  the  planet  through  its  node.  It  was  thus 
that  Lalande  determined  the  elements  of  the  orbit  of  Uranus, 
with  a  near  approximation  to  the  truth,  within  one  year  of  the 
period  of  the  first  discovery  of  that  planet  by  Sir  W.  Herschel. 

Mean  Elements  and  their  Variations. 

232.  The  elements  of  the  planetary  orbits,  obtained  by  the 
foregoing  processes,  are  the  true  elements  at  the  periods  when 
the  observations  are  made.  Upon  determining  them  at  different 
periods,  it  appears  that  they  are  subject  to  minute  variations, 
A  comparison  of  the  values  found  at  various  distant  epochs 


VARIATIONS    OF    THE    ELEMENTS.  '99 

shows  that  they  are  slowly  changing  from  century  to  century, 
and  that  the  changes  experienced  during  equal  long  periods  of 
time  are  very  nearly  the  same.  The  amount  of  the  variation  of 
an  element  in  a  period  of  100  years  is  called  its  Secular  Varia- 
tion. Upon  reducing  the  elements,  found  at  different  times,  to 
the  same  epoch,  by  allowing  for  the  proportional  parts  of  the 
secular  variations,  the  different  results  for  each  element  are 
found  to  differ  slightly  from  each  other,  which  shows  that 
the  elements  are  also  subject  to  slight  periodical  variations. 
These  variations  being  very  minute,  the  true  elements  can 
never  differ  much  from  the  mean,  or  those  from  which  they  de- 
viate periodically  and  equally  on  both  sides. 

The  mean  elements  at  an  assigned  epoch  may  be  had  by  find- 
ing: the  true  elements  at  various  times,  and  reducino:  them  to  the 
given  epoch,  by  making  allowance  for  the  proportional  parts  of 
the  secular  variations,  and  then  taking  for  each  element  the 
mean  of  all  the  particular  values  obtained  for  it. 

233.  A  comparison  of  the  mean  values  of  the  same  element, 
found  at  distant  epochs,  makes  known  the  variation  of  its  mean 
value  in  the  interval  between  them,  from  which  the  secular 
variation  may  be  deduced  by  simple  proportion. 

234.  The  elements  of  the  moon's  orbit  are  also  subject  to  con- 
tinual variations.  These  are,  for  the  most  part,  periodic,  and 
are  far  greater  than  the  variations  of  the  corresponding  elements 
of  a  planet's  orbit.  It  will  be  seen  then,  that  in  determining  the 
mean  elements,  a  much  greater  number  of  observations  will  be 
required  than  in  the  case  of  a  planetary  orbit.  The  mean  node 
and  perigee  have  a  rapid  and  nearly  uniform  progressive  mo- 
tion. Theory  shows  that  the  other  mean  elements,  with  the 
exception  of  the  semi-major  axis,  are  subject  to  secular  varia- 
tions, but  their  effect  has  hitherto  been  very  inconsiderable. 

235.  The  mean  elements  which  have  been  derived  as  above 
directly  from  observation,  have  subsequently  been  verified  and 
corrected,  by  comparing  the  computed  with  the  observed  places 
of  the  p'anet ;  and  for  this  purpose  many  thousands  of  observa- 
tions have  been  made. 

236.  Tables  II  and  III  contain  the  elements  of  the  orbits  of 
the  principal  planets,  and  of  the  moon's  orbit,  together  with 
their  secular  variations,  for  the  beginning  of  the  year  1801 ;  and 


100  ASTRONOMY. 

also  the  elements  of  the  orbits  of  the  four  small  planets,  Vesta, 
Juno,  Ceres  and  Pallas,  for  the  beginning  of  the  year  1820. 

If  an  element  be  desired  for  any  time  different  from  the  epoch 
of  the  table,  we  have  only  to  allow  for  the  proportional  part  of 
the  secular  variation  in  the  interval  between  the  given  time  and 
the  epoch  of  the  table. 

237.  It  will  be  seen,  on  inspecting  Table  II,  that  the  mean 
distances  of  the  planets  from  the  sun,  or  the  semi-major  axes  of 
their  orbits,  are  the  only  elements  that  are  invariable.  The  rest 
are  subject  to  minute  secular  variations.  The  nodes  have  all 
retroorade  motions.  The  perihelia,  on  the  contrary,  have  direct 
motions,  Avith  the  single  exception  of  the  perihelion  of  the  orbit 
of  Venus,  which  has  a  retrograde  motion.  The  eccentricities  of 
some  of  the  orbits  are  increasing,  of  others  diminishing.  That 
of  the  earth's  orbit  is  diminishing. 

The  node  of  the  moon's  orbit  has  a  retrograde  motion,  and  the 
perihelion  a  direct  motion.  The  former  accomplishes  a  tropical 
revolution  in  6788.50982  days,  or  about  18  years  214  days; 
and  the  latter  in  3231.4751  days,  or  in  about  8  years  309  days. 
The  mean  motion  of  the  node,  and  the  mean  motion  of  the 
perigee,  are  both  subject  to  a  slow  secular  diminution. 


CHAPTER     IX. 

of  the  determination  op  the  place  of  a  planet,  or 
of  the  sun,  or  moon,  for  a  given  time,  by  the 
elliptical  theory  ;  and  of  the  verification  of 
Kepler's  laws. 

Place  of  a  Planet^  or  of  the  Sun,  or  Moon,  in  its  Orbit. 
238.  The  angle  contained  between  the  line  of  apsides  of  a 
planet's  orbit  and  the  radius  vector,  as  reckoned  from  the  peri- 
helion towards  the  east,  is  called  the  True  Aiiomaly.     Thus, 
let  B  P  A  P'  (Fig.  45)  represent  the  orbit,  B  the  perihelion,  and 


PLACE  OP  A  PLANET  IN  ITS  ORBIT.  101 

P  the  position  of  the  planet ;  then,  B  S  P  is  its  true  anomaly. 
The  angle  contained  between  the  line  of  apsides  and  the  mean 
place  of  the  planet,  also  reckoned  from  the  perihelion  towards 
the  east,  is  called  the  Mean  Ajiomaly.  Thus,  let  M  be  the 
mean  place  of  a  planet  at  the  time  P  is  its  true  place,  and  B  S  M 
will  be  its  mean  anomaly. 

Describe  a  circle  B  />  A  on  the  line  of  apsides  as  a  diameter  ; 
through  P  draw  ;>  P  D  perpendicular  to  the  line  of  apsides,  and 
join  j>  and  C  :  the  angle  B  C  jj,  which  the  line  thus  determined 
makes  with  the  line  of  apsides,  is  called  the  Eccentric  Ano- 
maly. 

The  corresponding  angles  appertaining  to  the  sun's  apparent 
orbit,  and  to  the  moon's  orbit,  have  received  the  same  appel- 
lations. 

239.  The  interval  between  two  consecutive  returns  of  a  body 
to  either  apsis  of  its  orbit,  is  called  the  Anomalistic  Revolution. 
The  anomalistic  revolution  of  the  earth,  or  of  the  sun  in  its  ap- 
parent orbit,  is  termed  also  the  Anomalistic  Year. 

240.  The  periodic  time,  or  the  mean  motion  of  a  body,  and 
the  motion  of  the  apsis  of  its  orbit,  being  known,  the  anomalistic 
revolution  may  be  easily  computed.  Let  m  —  the  sidereal  mo- 
tion of  the  apsis  answering  to  the  periodic  time,  and  M  =  the 
mean  daily  motion  of  the  planet ;  then, 

M  :  1  d. : :  m  :  a:  =  diff.  of  anomalistic  rev.  and  periodic  time. 

241.  When  the  epoch  of  any  one  passage  of  a  planet  through 
its  perihelion,  or  of  the  sun  or  moon  through  its  perigee,  has 
been  found,  we  may,  by  means  of  the  anomalistic  revolution, 
deduce  from  it  the  epoch  of  every  other  passage. 

242.  The  length  of  the  anomalistic  year  exceeds  that  of  the 
sidereal  year  by  4  m.  43.9  s. 

243.  From  the  anomalistic  revolution,  and  the  epoch  of  the 
last  passage  through  the  perihelion  or  perigee  (as  the  case  may 
be),  we  may  derive  the  mean  anomaly  for  any  given  time.  Let 
T  =  the  anomalistic  revolution,  t  =  the  time  that  has  elapsed 
since  the  last  passage  through  the  perihelion  or  perigee,  and  A 
=  the  mean  anomaly  :  then, 

T  :  360°  •.-.t:  A  =360°  L  .  .  .  (60). 

244.  The  place  of  a  body  in  its  elliptical  orbit  is  ascertained 


102  ASTRONOMY. 

by  finding  its  true  anomaly.  The  problem  which  has  for  its 
object  the  determination  of  the  true  anomaly  from  the  mean, 
was  first  resolved  by  Kepler,  and  is  called  Kepler's  Problem. 
The  solution  of  it  may  be  found  ni  the  Appendix.  Another  and 
more  convenient  method  of  obtaining  the  true  anomaly,  is  to 
compute  the  equation  of  the  centre  from  the  mean  anomaly,  and 
add  it  to  the  mean  anomaly,  or  subtract  it  from  it,  according  to 
the  position  of  the  body  in  its  orbit  (Art.  185). 

Heliocentric  Place  of  a  Planet. 

245.  The  place  of  a  planet  in  the  plane  of  its  orbit  is  designa- 
ted by  its  orbit  longitude,  and  radius  vector.  To  find  the  orbit 
longitude  we  have  the  equation, 

long.  =  long,  of  perihelion  +  true  anomaly. 

The  orbit  longitude  may  also  be  deduced  from  the  mean  lon- 
gitude, by  adding  or  subtracting  the  equation  of  the  centre. 

The  radius  vector  results  from  the  polar  equation  of  the  ellip- 
tic orbit  (Art.  227),  viz : 

V  =  ^(^"~^')  .  .  .  (61). 
1  -f  e  cos  X 

in  which  x  denotes  the  true  anomaly,  e  the  eccentricity,  and  a 

the  semi-major  axis. 

246.  Now  to  find  the  heliocentric  longitude  and  latitude, 
which  ascertain  the  position  of  the  planet  with  respect  to  the 
ecliptic,  the  triangle  N  P  p  (Fig.  41)  gives, 

sin  P  />  =  sin  N  P  sin  P  N  p  ; 
or,  sin  lat.  =  sin  (orbit  long.  —  long,  of  node)  x  sin  (inclin.) .  .  (62) ; 
and 

cos  P  N  p  =  tang  N  /?  cot  N  P,  or  tang  N  p  =  tang  N  P  cos  P  N  2? , 
or, 

tang  (long.  —  long,  of  node)  =  tang  (orbit  long.  —  long,  of  node) 
X  cos  (inclination)  .  .  .  (63). 

Geocentric  Place  of  a  Planet. 

247.  From  the  heliocentric  longitude  and  latitude  and  the 
radius  vector  of  a  planet,  to  find  the  geocentric  Imigitude  and 
latitude.  Let  S  (Fig,  41)  be  the  sun,  E  the  earth,  P  the  planet, 
*  its  reduced  place,  and  V  the  vernal  equinox.  Denote  the  he- 
liocentric longitude  V  S  *  by  L,  the  heliocentric  latitude  P  S  * 
by  It  and  the  radius  vector  S  P  by  v  ;  and  denote  the  geocentric 


GEOCENTRIC    PLACE    OF    A    PLANET. 


103 


longitude  by  G,  and  the  geocentric  latitude  by  X.  Also  let  E  = 
S  E  'ff  the  elongation  ;  C  =  E  S  *  the  commutation  ;  A  =  S  *  E  the 
annual  parallax  ;  and  r  =  S  E  the  radius  vector  of  the  earth. 

Now, 

VE*-SE*  +  VES, 

or,  G  =  E  +  long,  of  sun. 

This  equation  will  make  known  the  geocentric  longitude, 
when  the  value  of  E  is  found.  In  the  triangle  S  E  *,  the  side  S 
*  =  S  P  cos  P  S  *  =  ?;  cos  I,  and  is  therefore  known,  the  side  E  S 
is  given  by  the  elliptical  theory  (Art.  245),  and  the  angle  C  may 
be  derived  from  the  following  equation  :  C  =  VSE  —  "^8*  = 
long,  of  earth  —  long,  of  planet :  and  to  find  E  we  have,  by 
Trigonometry, 
E  S  +S  * :  E  S— S  * : :  tan  i  (E  *  S +S  E  *) :  tan  1  (E *  S— SE*), 

or,  r  +  V  cos  ^  :  r  —  v  cos  I :  :  tang  ^  (A  +  E) :  tang  ^  (A  —  E) ; 


whence,      tang  ^  (A  —  E)  = 


V  cos  I 


1  — 


V   COS  I 

V  cost 


tangi(A  +  E); 


-J    ,    v  cos  I 


tangi(A  +  E). 


Let  tang 


V  cos  / 


tang  |(A  —  E)  = 


Then, 
1  — 


1  + 


tang  I  (A +  E); 


or,    tang  ^  (A  —  E)  =  tang  (45°  —  6)  tang |(A  4-  E)  .  .  .  (64). 
But,     A  +  E  =  180°  —  C,  and  E  =  ^A  +  E)  —  1  (A  —  E). 
Next,  to  find  the  geocentric  laitude. 

S  *  tang  ^  =  P  *  =  E  'TT  tang  X, 

S  *  _  tang  X 

E^ 


whence, 

but, 

and  therefore 

or, 


tanof  Z 


S  *  :  E  ^  : :  sin  E  :  sin  C,  or  ^  =  !Hi^  , 
'       ET      sin  C ' 


sin  E      tano-x 


sin  C      tangZ  ' 

.        .      sin  E  tangZ  ,rr'\ 

tang  X  = -. — _o_ .  .  .  (65). 

^              sm  C  ^     ' 


104  ASTRONOMY. 

248.  When  a  planet  is  in  conjunction  or  opposition,  the  sines 
of  the  angles  of  elongation  and  commutation  are  each  nothing. 
In  these  cases,  then,  the  geocentric  latitude  cannot  be  found  by 
the  preceding  formula,  it  may  however  be  easily  determined  in  a 
different  manner.  Suppose  the  planet  to  be  in  conjunction  at  P 
(Fig.  42) ;  then, 

^      P  -^  P*        . 

but  the  triangle  S  P  -jt  gives, 

P  "jr  =  v  sin  Z,  and  S'!f  =  v  cos  I ;  and  E  S  =  r  ; 

hence,  tangX  = ,  .  .  (66).* 

249.  To  find  the  distance  of  the  planet  from  the  earth,  repre- 
sent the  distance  by  D  ;  then,  from  the  triangles  S  P  *  and  E  P 
tf  (Fig.  41),  we  have, 

P  "T  =  E  P  sin  P  E  cr  =  D  sin  X, 

and  P  -r  =  S  P  sin  P  S  *  =  iJ  sin  Z ; 

whence,  D  =  —. .  .  .  (67). 

'  sui  X  ^     ' 

250.  The  distance  of  a  planet  being  known,  its  horizontal 
parallax  may  be  computed  from  the  equation 

sin  H  =  ^  .  .  .  (68).      (Art.  98). 

Places  of  the  iSun  and  Moon. 

251.  The  place  of  the  sun,  as  seen  from  the  earth,  may  be 
easily  deduced  from  the  heliocentric  place  of  the  earth  ;  for,  the 
longitude  of  the  sun  is  equal  to  the  heliocentric  longitude  of  the 
earth  plus  180°,  and  the  radius  vector  of  the  earth's  orbit  is  the 
same  as  the  distance  of  the  sun  from  the  earth.  But  it  is  more 
convenient  to  regard  the  sun  as  describing  an  orbit  around  the 
earth,  and  to  compute  its  true  anomaly,  (Art.  244),  and  thence 
the  longitude  and  radius  vector  by  the  equation 

long.  =  true  anomaly  +  long,  of  perigee, 
and  the  polar  equation  of  the  orbit. 

252.  The  orbit  longitude  and  the  radius  vector  of  the  moon 


*  For  opposition  and  inferior  conjunction,  the  sign  of  cos  I  must  be  changed. 


VERIFICATION    OF    KEPLER's    LAWS.  105 

are  found  by  the  same  process  as  the  longitude  and  radius  vec- 
tor of  the  sun.  The  orbit  longitude  being  known,  the  ecliptic 
longitude  and  the  latitude  may  be  determined  by  a  process  pre- 
cisely similar  to  that  by  which  the  heliocentric  longitude  and 
latitude  of  a  planet  are  found  (Art.  245). 

Verijicatmi  of  Keplefs  Laws. 

253.  If  Kepler's  first  two  laws  be  true,  then  the  geocentric 
places  of  the  planets,  computed  by  the  process  that  we  have 
described  (Art.  246),  which  is  founded  upon  them,  ought  to 
agree  with  the  true  geocentric  places  as  obtained  for  the  same 
time  by  direct  observation  :  or,  the  heliocentric  places  computed 
from  the  observed  geocentric  places  (Art.  223),  ought  to  agree 
with  the  same  as  computed  by  the  elliptic  theory  (Arts.  245,  246). 
Now,  a  great  number  of  comparisons  have  been  made  between 
the  observed  and  computed  places,  and  in  every  instance  a  close 
agreement  between  the  two  has  been  found  to  subsist.  We 
infer,  therefore,  that  the  motions  of  the  planets  must  be  very 
nearly  in  conformity  with  these  laws. 

The  truth  of  the  third  law  has  been  established  by  a  direct 
comparison  of  the  mean  distances  of  the  different  planets  with 
their  periodic  times. 

254.  Kepler's  laws  have  been  verified  for  the  sun  and  moon, 
in  a  similar  manner. 

255.  The  relative  distances  of  the  sun,  or  moon,  at  different 
times,  result  from  observations  upon  the  apparent  diameter^ 
upon  the  principle  that  any  two  distances  are  inversely  propor- 
tional to  the  corresponding  apparent  diameters.  Let  A  =  semi- 
diameter  corresponding  to  the  mean  distance,  and  h  —  semi- 
diameter,  corresponding  to  any  distance  D  :  then 

(5 :  A  : :  1  :  D  ;  whence,  D  =  —    ...  (69) ; 

an  equation  which,  when  A  has  been  found,  will  make  known 
the  distance  corresponding  to  any  observed  semi-diameter  5,  in 
terms  of  the  mean  distance  as  a  unit. 

Now,  to  find  A,  denote  the  greatest  and  least  semi-diameters 
respectively  by  <5',  5",  and  the  corresponding  distances  by  D'  and 
D"j  and  we  have, 


5' '  6" 


14 


106  ASTRONOMY. 

andthcnce,       i  (D' +  D")  ^  ^^+|,)  ^''^^Ky+-^)  ' 

2  5'  6" 
whence,  A  =  —^  .  .  .  (70). 

256.  The  distance  of  the  sun  or  moon  in  terms  of  the  mean 
distance  as  a  unit,  may  be  found  in  a  similar  manner ;  but  it 
may  be  had  more  accurately  by  means  of  a  principle  which  has 
been  discovered  from  observation,  namely,  that  the  distance  is 
inversely  proportional  to  the  square  root  of  the  daily  angular 
motion. 


CHAPTER   X. 

OF  THE  INEaUALITIES  OF  THE  MOTIONS  OF  THE  PLANETS 
AND  OF  THE  MOON  ;  AND  OF  THE  CONSTRUCTION  OF  TA- 
BLES   FOR    FINDING    THE    PLACES    OF    THESE    BODIES. 

257.  It  is  a  general  law  of  nature,  discovered  by  Sir  Isaac 
Newton,  that  bodies  tend,  or  gravitate  towards  each  other,  with 
a  force  directly  proportional  to  their  masses  and  inversely  pro- 
portional to  the  square  of  their  distance.  The  force  which 
causes  one  body  to  gravitate  towards  another,  is  supposed  to 
arise  from  a  mutual  attraction  existing  between  the  particles  of 
the  two  bodies,  and  is  hence  called  the  Attraction  of  Gravita- 
tion. This  force  of  attraction,  common  to  all  the  bodies  of  the 
Solar  System,  is  the  general  physical  cause  of  their  motions. 
The  sun's  attraction  retains  the  planets  in  their  orbits,  and  the 
planets  by  their  mutual  attractions  slightly  alter  each  other's 
motions.  The  reasoning  by  which  Newton^s  Theory  of  Uni- 
versal Gravitation  is  established,  appertains  to  Physical  Astron- 
omy, and  will  be  presented  in  another  part  of  the  work. 

258.  If  a  planet  were  acted  on  by  no  other  force  than  the 
attraction  of  the  sun,  it  is  proved  that  its  orbit  would  be  accu- 


INEaUALITIES    OF    THE    PLANETARY    MOTIONS.  107 

rately  an  ellipse,  and  that  the  areas  described  by  its  radius  vector 
in  equal  times,  would  be  precisely  equal.  But,  it  is  in  reality 
attracted  by  the  other  planets,  as  well  as  the  sun,  and  therefore 
its  actual  motions  cannot  be  in  strict  conformity  with  the  laws 
of  Kepler.  In  fact,  if  we  descend  to  great  accuracy,  the  agree- 
ment between  the  observed  and  computed  places  noticed  in  Art. 
253,  is  found  not  to  be  exact.  The  deviations  from  the  elliptic 
motion  which  are  produced  by  the  attractions  of  the  planets, 
are  called  Perturbations^  or,  in  Plane  Astronomy,  Inequalities. 
Although,  as  we  have  just  seen,  the  fact  of  the  existence  of  ine- 
qualities in  the  motions  of  the  planets  is  discoverable  from  ob- 
servation, their  laws  cannot  be  determined  without  the  aid  of 
theory. 

259.  In  treating  of  the  perturbations  in  the  motions  of  one 
planet,  resulting  from  the  attractions  of  another,  the  attracting 
planet  is  called  the  Disturbing  Body^  and  the  force  which  pro- 
duces the  perturbations  the  Disturbing  Force.  To  find  the 
disturbing  force,  let  P  (Fig.  46)  be  the  planet,  S  the  sun,  and 
M  the  disturbing  body ;  and  let  P  D  represent  the  attraction  of 
M  for  the  planet.  Decompose  P  D  into  two  forces,  P  E  and  P  F, 
one  of  which,  P  E,  is  equal  and  parallel  to  S  G,  the  attraction  of 
M  for  the  sun ;  the  other,  P  F,  will  be  known  in  position  and 
intensity.  The  two  forces,  P  E  and  S  G,  being  equal  and  paral- 
lel, they  cannot  alter  the  relative  motion  of  the  sun  and  planet, 
and  accordingly  may  be  left  out  of  account :  there  remains, 
therefore,  the  component  P  F,  which  will  be  wholly  effectual  in 
disturbing  this  motion.     This,  then,  is  the  disturbing  force. 

It  happens  in  the  case  of  each  planet,  that  the  distances  of  some 
of  the  other  planets  are  so  great,  that  their  disturbing  forces  are 
insensible.  The  attractions  of  these  bodies  for  the  sun  and 
planet  are  sensibly  equal  and  parallel.  Owing  to  the  great  dis- 
tance of  the  planets  from  each  other,  and  the  smallness  of  their 
mass  compared  with  that  of  the  sun,  the  disturbing  force  is  in 
every  instance  very  minute  in  comparison  with  the  sun's  at- 
traction. 

260.  It  is  plain  that  the  disturbing  force  will,  in  general, 
be  obliquely  inclined  to  the  perpendicular  to  the  plane  of 
the  orbit,  P  K  ;  the  tangent  to  the  orbit,  P  T  ;  and  the  radius 
vector,  P  S ;   and  may,  therefore,  be  decomposed  into  forces 


108  ASTRONOMY. 


acting  along  these  lines.  The  component  along  the  perpendi- 
cular will  alter  the  latitude,  and  the  two  others  both  the  longi- 
tude and  radius  vector  ;  that  along  the  tangent  by  changing  the 
velocity  of  the  planet;  and  that  along  the  radius  vector  by 
changing  the  gravity  towards  the  sun.  It  appears,  therefore, 
that  the  disturbing  force  produces  at  the  same  time  perturbations 
or  inequalities  of  longitude,  of  latitude,  and  of  radius  vector. 

261.  Let  us  now  consider  how  these  inequalities  may  be 
determined.  And,  in  the  first  place,  the  inequalities  produced 
by  each  disturbing  body  may  be  separately  investigated  upon 
mechanical  principles,  as  if  the  other  bodies  did  not  exist,  for 
the  reason  that  the  effect  of  each  disturbing  body  is  sensibly  the 
same  that  it  would  be  if  the  other  bodies  did  not  act.  That  this 
is  very  nearly,  if  not  quite  true,  may  be  at  once  inferred  from 
the  minuteness  of  the  whole  disturbance  produced  by  the  joint 
action  of  all  the  disturbing  forces  of  the  system.  The  problem 
which  has  for  its  object  the  determination  of  the  inequalities  in 
the  motions  of  one  body,  in  its  revolution  around  a  second,  pro- 
duced by  the  attraction  of  a  third,  is  called  the  Problem  of  the 
Three  Bodies.  If,  in  the  case  of  any  one  planet,  this  problem 
be  resolved  for  each  of  the  other  bodies  of  the  system  which 
occasion  sensible  perturbations,  all  the  inequalities  to  which  the 
motion  of  the  planet  is  subject  will  become  known. 

262.  The  general  solution  of  the  problem  of  the  three  bodies, 
that  is  for  any  mass  and  distance  of  the  disturbing  body,  or  any 
intensity  of  the  disturbing  force,  cannot  be  effected  in  the  exist- 
ing state  of  the  mathematical  sciences.  But  the  problem  has 
been  resolved  for  the  case  that  presents  itself  in  nature,  in  which 
the  disturbing  force  is  very  minute  in  comparison  with  the  cen- 
tral attraction.  The  results  obtained  by  the  analysis,  are  certain 
analytical  expressions  for  the  perturbations  in  longitude,  latitude, 
and  radius  vector,  involving  variables  and  constants. 

263.  The  general  expression  for  the  whole  perturbation  in 
longitude,  due  to  the  action  of  any  one  disturbing  body,  is 

C  sin  (P'  —  P)  +  C  sin  2  (P'  —  P)  +  C"  sin  3  (P'  —  P)  +  &c.  (71), 

in  which  C,  C,  &c.  are  constants,  P  the  heliocentric  longitude 
of  the  body  disturbed,  and  P'  that  of  the  disturbing  body.  The 
number  of  terms  is,  strictly  speaking,  indefinite,  but  they  form  a 


DETERMINATION    OP    INEaUALITIES.  109 

decreasiiisr  series  :  and  the  value  of  the  first  term  never  amounts 
to  more  than  a  few  seconds  ;  so  that  only  a  small  number  of  the 
first  terms  (which  will  be  difierent  in  different  cases)  need  to  be 
used. 

264.  The  constants  C,  C,  &,c.  are  to  be  determined  from  ob- 
servation ;  they  may,  however,  be  determined  in  the  case  of 
some  of  the  planets  from  theory  alone.  The  process  of  finding 
them  from  observation  is  as  follows  :  Suppose  that  the  earth  is 
the  body  whose  perturbations  are  under  consideration,  and  let  D 
denote  the  perturbation  in  longitude,  produced  by  the  joint  ac- 
tion of  all  the  disturbing  forces.  Then,  supposing,  for  the  sake 
of  simplicity,  that  the  expression  for  the  perturbation  due  to 
each  disturbing  body,  consists  of  but  two  terms,  we  have, 

D  =  C  sin  (P'  —  P)  +  C  sin  2  (P'  —  P)  +  c  sin  (P"  —  P)  -j-  c'  sin 

2(P"  — P)+&c.  .  .  .  (72). 
Find,  by  observation,  the  heliocentric  longitude  of  the  earth, 
and  take  the  difference  between  this  and  the  longitude  as  com- 
puted for  the  same  time  by  the  elliptical  theory.  This  differ- 
ence will  be  the  value  of  D  at  the  time  of  the  observation.  P, 
P',  P",  &c.  the  heliocentric  longitudes  of  the  earth  and  of  the 
disturbing  bodies,  and  consequently  P'  —  P,  P"  —  P,  &c,  are 
given  by  the  elliptical  theory.  Thus,  in  the  above  equation  all 
will  be  known  but  C,  C,  c,  c',  &c.  By  repetitions  of  this  process, 
as  many  equations  may  be  obtained  as  there  are  constants  to  be 
determined,  and  from  these  the  values  of  the  constants  may  be 
computed.  It  is  usual,  however,  to  obtain  a  much  greater  num- 
ber of  equations  than  there  are  constants  ;  as,  by  combining  them 
according  to  certain  rules,  much  more  exact  values  of  the  con- 
stants may  be  derived. 

265.  In  the  expression, 

C  sin  (P'  _  P)  -f-  C  sin  2  (P'  —  P)  +  &c., 
for  the  perturbation  in  longitude,  due  to  the  action  of  a  disturb- 
ing  body,  each  term,  C  sin  (P' —  P),  C  sin  2(P'  — P),  (fcc,  is 
technically  termed  an  Equation,  and  is  considered  as  represent- 
ing a  specific  inequality.  The  angle  P'  —  P,  or  2  (P'  —  P),  or 
other  multiple  of  P' —  P,  the  sine  of  which  enters  into  the  equa- 
tion of  an  inequality,  is  called  the  Argument  of  the  inequality  ; 
and  the  constant  is  called  the  Coefficient  of  the  inequality.     As 


110  ASTRONOMY. 

the  greatest  value  of  the  sine  of  the  ars^ument  is  unity,  the  co- 
efficient is  equal  to  the  greatest  value  of  the  inequality. 

266.  The  coefficient  being  known,  the  value  of  the  inequality 
at  any  particular  time  will  become  known,  if  that  of  the  argu- 
ment be  found.  Now,  the  argument  is  the  difference  between 
the  longitudes  of  the  disturbing  body  and  disturbed  body,  or 
some  multiple  of  this  difference,  and  may  be  found  by  the  ellip- 
tical theory.  In  practice,  the  mean  longitudes  may  be  taken, 
without  material  error,  in  place  of  the  true,  and  these  are  easily 
deduced  from  the  mean  longitudes  at  a  given  epoch,  by  means  of 
the  mean  motions  in  longitude  of  the  two  bodies.  When  the 
values  of  all  the  inequalities  in  longitude  have  been  separately 
determined,  by  taking  their  algebraic  sum,  we  shall  have  the 
correction  to  be  applied  to  the  elliptic  longitude,  in  order  to  find 
the  exact  longitude. 

267.  The  general  expression  for  the  total  perturbation  of  ra- 
dius vector,  due  to  the  action  of  one  body,  is 

C  cos  (P'  —  P)  +  C  cos  2  (  P'— P)  -f  C"  cos  3  (P'— P)  -f  &c.  (73). 

As  in  the  expression  for  the  perturbation  of  longitude,  each  term 
is  called  an  equation,  and  represents  a  distinct  inequality,  the 
constant  being  the  coefficient,  and  the  variable  angle,  the  cosine 
of  which  enters  into  the  equation,  the  argument  of  the  inequality. 
The  amounts  of  the  different  inequalities,  at  an  assumed  time, 
are  computed  after  the  same  manner  as  those  of  the  inequalities 
of  longitude,  and  being  added  together  with  thtir  algebraical 
signs,  will  give  the  correction  to  be  applied  to  the  elliptic  radius 
vector. 

268.  The  perturbation  in  latitude  is  very  minute.  The  ine- 
qualities of  latitude,  as  of  longitude  and  radius  vector,  are  repre- 
sented by  equations,  composed  of  a  constant  coefficient  and  the 
sine  or  cosine  of  a  variable  argument,  or  of  the  form  C  sin  A  or 
C  cos  A. 

269.  The  arguments  of  the  inequalities  we  have  been  con- 
sidering, are  angles  depending  upon  the  configurations  of  the  dis- 
turbing and  disturbed  planets  with  respect  to  each  other  and  the 
sun,  and  also,  in  some  cases,  with  respect  to  the  nodes  and  peri- 
helia of  their  orbits.  Whenever  these  configurations  become 
the  same,  as  they  will  periodically,  the  arguments,  and  therefore 


PERIODIC    AND    SECULAR    INECIUALITIES.  Ill 

the  inequalities  themselves,  will  have  the  same  value.     It  fol- 
lows, therefore,  that  the  inequalities  in  question  are  periodic. 

The  interval  of  time  in  which  an  inequality  passes  through  all 
its  gradations  of  positive  and  negative  value,  is  called  the  Period 
of  the  inequality.  It  is  manifestly  equal  to  the  interval  of  time 
employed  by  the  argument  in  increasing  from  zero  to  360°  ;  for, 
in  this  interval  sin  A  or  cos  A  takes  all  its  values,  both  positive 
and  negative,  and  at  the  expiration  of  it  recovers  the  same  value 
again. 

270.  It  has  been  stated,  that  the  elements  of  the  elliptic  orbits 
of  the  planets  are,  for  the  most  part,  subject  to  a  slow  variation 
from  century  to  century.  Investigations  in  Physical  Astronomy 
have  established  that  the  variations  of  the  elements  are  due  to 
the  action  of  the  disturbing  forces  of  the  planets,  and  that  they 
are  not  progressive  (except  in  the  cases  of  the  longitude  of  the 
node  and  the  longitude  of  the  perihelion),  but  are  really  periodic 
inequalities,  whose  periods  comprise  many  centuries.  From  the 
great  lengths  of  their  periods  these  inequalities  are  termed  *S'ecM- 
lar  Inequalities,  in  order  to  distinguish  them  from  the  inequali- 
ties of  the  elliptic  motion,  denominated  Periodic  Inequalities, 
the  periods  of  which  are  comparatively  short. 

Physical  Astronomy  ftirnishes  expressions  called  Secular 
Equations,  which  give  the  value  of  an  element  at  any  assumed 
time. 

271.  The  inequalities  of  the  moon's  motions  arise  from  the 
disturbing  action  of  the  sun.  The  attractions  of  the  planets  for 
the  moon  and  earth  are  sensibly  equal  and  parallel.  The  lunar 
inequalities  are  investigated  upon  the  same  principles  as  the 
planetary,  and  are  represented  by  equations  of  the  same  general 
form,  that  is,  consisting  of  a  constant  coefficient  and  the  sine  or 
cosine  of  a  variable  argument.  They  far  exceed  in  number  and 
magnitude  those  of  any  single  planet. 

272.  There  are  three  lunar  inequalities  of  longitude  which 
are  prominent  above  the  rest,  and  were  early  discovered  by 
observation. 

The  most  considerable  is  called  the  Ejection,  and  was  disco- 
vered by  Ptolemy  in  the  first  century  of  the  Christian  era.  It 
has  for  its  argument,  double  the  angular  distance  of  the  moon 


112  ASTRONOMY. 

from  the  sun  minus  the  mean  anomaly  of  the  moon,  and  amounts 
when  greatest  to  1°  20'  30." 

The  second  is  called  the  Variation,  and  was  discovered  in 
the  sixteenth  century  by  Tycho  Brahe.  Its  argument  is  double 
the  angular  distance  of  the  moon  from  the  sun,  and  its  maximum 
value  is  35'  42." 

The  third  is  denominated  the  Annual  Equation,  from  the 
circumstance  of  its  period  being  an  anomalistic  year.  Its  argu- 
ment is  the  mean  anomaly  of  the  sun.  When  greatest,  it 
amounts  to  11'  12". 

273.  The  discovery  of  the  other  lunar  inequalities  (with  the 
exception  of  one  inequality  of  latitude)  is  due  to  Physical 
Astronomy. 

The  whole  number  of  lunar  inequalities  of  longitude,  accord- 
ng  to  Burckhardt,  is  32. 

274.  To  present  now  at  one  view,  the  entire  process  of  finding 
the  exact  heliocentric  place  of  a  planet,  or  the  geocentric  place 
of  the  moon,  at  any  assumed  time. 

1.  Seek  the  elements  of  the  elliptic  orbit  from  a  table  of  ele- 
ments, such  as  Table  II  or  III,  allowing  for  the  proportional  part 
of  the  secular  variation,  or  (more  exactly)  obtain  them  from  their 
secular  equations  (Art.  270). 

2.  Compute  the  longitude,  latitude,  and  radius  vector,  by  the 
elliptic  theory  (Arts.  245,  246). 

3.  Compute  the  values  of  the  inequalities  in  longitude  and 
latitude  and  of  radius  vector  by  means  of  their  equations  (Art. 
266),  and  apply  them  individually  with  their  proper  signs,  as 
corrections  to  the  elliptic  values  of  the  longitude,  latitude  and 
radius  vector. 

275.  If  we  suppose  the  sun  to  be  in  motion,  instead  of  the 
earth,  its  inequalities  will  be  the  same  as  those  to  which  the 
motion  of  the  earth  is  actually  subject. 

276.  When  the  heliocentric  place  of  a  planet  has  been  found, 
its  geocentric  place,  if  required,  may  be  determined  by  the  pro- 
cess explained  in  Art.  247. 

Construction  of  Tables. 

277.  The  determination  of  the  place  of  the  sun  or  moon,  or 
of  a  planet,  may  be  greatly  facilitated  by  the  use  of  tables. 
The  principles  and  modes  of  construction  of  tables  adapted  to 


TABLES    FOR    THE    SUn's    LONGITUDE.  113 

this  purpose  are  nearly  the  same  for  each  body.  We  will  first 
explain  the  mode  of  constructing  tables  for  facilitating  the  com- 
putation of  the  sun's  longitude.     We  have  the  equation. 

True  long.  =  mean  long.  -\-  equa.  of  centre  +  inequalities  + 
nutation. 

If,  then,  tables  can  be  constructed  that  will  furnish  by  inspec- 
tion the  mean  longitude,  the  equation  of  the  centre,  the  amounts 
of  the  various  inequalities  in  longitude,  and  the  nutation  in 
longitude,  at  any  assumed  time,  we  may  easily  find  the  true 
longitude  at  the  same  time. 

278.  1.  For  the  mean  longitude.  The  sun's  mean  motion  in 
longitude  in  a  mean  tropical  year,  is  360°.  From  this  we  may 
find  by  proportion,  the  mean  motions  in  a  common  year  of  365 
days  and  a  bissextile  year  of  366  days. 

With  these  results  and  the  mean  longitude  for  the  epoch  of 
Jan.  1,  1801,  we  may  easily  derive  the  mean  longitude  at  the 
beginning  of  each  of  the  years  prior  and  subsequent  to  the  year 
1801.  The  second  column  of  Table  XVIII  contains  the  mean 
longitude  of  the  sun  at  the  beginning  of  each  of  the  years 
inserted  in  the  first  column.  The  third  column  of  this  table 
contains  the  mean  longitude  of  the  perigee  at  the  same  epochs : 
it  was  constructed  by  means  of  the  mean  longitude  of  the  peri- 
gee found  for  the  beginning  of  the  year  1800,  and  its  mean 
yearly  motion  in  longitude,  which  is  61  ".52.* 

Having  the  sun's  mean  daily  motion  in  longitude  (Art.  176), 
we  obtain  by  proportion  the  motion  in  any  proposed  number  of 
months,  days,  hours,  minutes,  or  seconds.  Table  XIX  contains 
the  respective  amounts  of  the  sun's  motion  from  the  commence- 
ment of  the  year  to  the  close  of  each  month  ;  Table  XX,  the 
sun's  mean  motion  for  days  from  1  to  31,  and  for  hours  from 
1  to  24 ;  and  Table  XXI,  the  same  for  minutes  and  seconds 
from  1  to  60.  With  these  tables,  the  sun's  mean  motion  in  lon- 
gitude in  the  interval  between  any  given  time  in  any  year  and 
the  beginning  of  the  year,  may  be  had  :  and  if  this  be  added  to 
the  epoch  for  the  given  year,  taken  out  from  Table  XVIII,  the 

*  The  quantities  in  Table  XVIII  are  called  Epochs,  The  Epoch  of  a  quantity- 
is  its  value  at  some  chosen  epoch. 

15 


114  ASTRONOMY. 

result  will  be  the  mean  longitude  at  any   given  time.     (See 
Problem  IX.) 

279.  Tables  XIX  and  XX  also  contain  the  motions  of  the 
sun's  perigee,  from  which  and  the  epoch  given  by  Table  XVIII, 
results  the  longitude  of  the  perigee  at  any  proposed  time.  The 
longitude  of  the  perigee  is  given  in  the  Solar  Tables,  for  the 
purpose  of  making  known  the  mean  anomaly,  the  mean  anom- 
aly being  equal  to  the  mean  longitude  minus  the  longitude  of 
the  perigee. 

280.  2.  For  the  equation  of  the  centre.  To  find  the  equation 
of  the  centre  of  an  orbit  we  have  the  following  equation  : 

Equa.  of  centre  =  A  sin  ^  +  B  sin  2  ^  -f-  C  sin  3  ^  +  &c. ; 

in  which  A,  B,  C,  <fcc.  are  constants  that  rapidly  decrease  in 
value,  and  which  may  be  determined  for  any  particular  orbit, 
and  &  the  mean  anomaly.  Now,  by  giving  to  the  mean  anomaly 
^  in  this  equation,  a  series  of  values  increasing  by  small  equal 
differences  (of  1°,  for  instance.)  from  zero  to  360°,  and  comput- 
ing the  corresponding  values  of  the  equation  of  the  centre ; 
then  registering  in  a  column  the  different  values  assigned  to  &, 
and  in  another  column  to  the  right  of  this,  the  computed  values 
of  the  equation  of  the  centre,  we  shall  obtain  a  table  which  will 
give  on  inspection  the  equation  of  the  centre  corresponding  to 
any  particular  mean  anomaly.  In  this  manner  was  constructed 
Table  XXV.  In  this  table,  however,  for  the  sake  of  compact- 
ness, the  values  of  the  equation,  instead  of  being  registered  in 
one  column,  are  put  in  as  many  different  columns  as  there  may 
be  different  numbers  of  signs  in  the  value  of  the  mean  anomaly  ; 
each  column  answering  to  the  particular  number  of  signs  placed 
at  the  head  of  it. 

If  the  equation  of  the  centre  at  an  assumed  time  be  required, 
find  the  mean  anomaly  by  the  tables  (Art.  279),  and  with  the  value 
found  for  it  take  out  the  equation  of  the  centre  from  Table 
XXV. 

The  given  quantity  with  which  a  quantity  is  taken  from  a 
table,  is  called  the  Argument.  Accordingly  the  mean  anomaly 
is  the  argument  of  the  equation  of  the  centre  in  Table  XXV. 

281.  3.  For  the  inequalities.  The  equations  of  the  inequali- 
ties, as  we  have  already  stated,  are  of  the  form  C  sin  A,  the 


TABLES    FOR    THE    SUN's    LONGITUDE.  115 

arsfiiment  A  beina^  the  difference  between  the  long-itude  of  the 
disturbing  planet  and  that  of  the  earth,  or  some  multiple  of  this 
difference.  With  the  equations  of  the  inequalities,  a  table  of 
each  inequality  may  be  constructed,  upon  the  same  principles  as 
Table  XXV.  But,  as  the  expression  for  the  whole  perturbation 
in  longitude  (Art.  263),  produced  by  any  one  planet,  involves 
only  two  variables,  the  longitude  of  the  earth  and  the  longitude 
of  the  planet,  it  is  thought  to  be  more  convenient  to  have  a  table 
of  double  entry.,  which  will  give  the  amount  of  the  perturbation 
by  means  of  the  two  variables  as  arguments.  Such  a  table  may 
be  constructed,  by  assigning  to  the  longitude  of  the  earth  and 
the  longitude  of  the  disturbing  planet  a  series  of  values  increas- 
ing by  a  common  difference,  and  computing  with  each  set  of  the 
values  of  these  quantities,  the  corresponding  amount  of  the 
perturbation. 

In  connection  with  the  tables  of  the  perturbations,  we  must  have 
tables  that  make  known  the  values  of  the  arguments  at  any  given 
time.  Now,  the  mean  longitude  of  the  sun  may  be  found  by 
the  solar  tables  (Art.  278),  and  thence  the  mean  heliocentric 
longitude  of  the  earth  by  subtracting  180°  ;  and  the  mean  longi- 
tude of  the  disturbing  planet  may  be  had  from  similar  tables. 
The  columns  of  Table  XVIII,  marked  I,  II,  III,  IV,  V,  VI,  VII, 
contain  the  arguments  of  all  the  perturbations,  for  the  beginning 
of  each  of  the  years  registered  in  the  first  column,  expressed  in 
thousandth  parts  of  a  circle.  Tables  XIX  and  XX  contain  the 
variations  of  the  arguments  for  months  and  hours.  Their  varia- 
tions for  minutes  and  seconds  are  too  small  to  be  taken,  into 
account.  With  these  tables  and  Table  XVIII,  the  values  of  the 
arguments  at  any  given  time  may  be  found,  and  by  means  of  the 
arguments  the  perturbations  may  be  taken  from  Tables  XXVIII, 
XXIX,  XXX,  XXXI,  XXXII,  and  XXXIII. 

282.  4.  For  the  nutation.  The  formula  for  the  lunar 
nutation  in  longitude,  is  17".3  sin  N,  —  0".2  sin  2  N,  in 
which  N  denotes  the  supplement  of  the  longitude  of  the 
moon's  ascending  node.  With  this  formula  the  second  col- 
umn of  Table  XXVII  was  constructed.  The  value  of  N, 
in  thousandth  parts  of  a  circle,  results  from  Tables  XVIII, 
XIX,  and  XX.  The  solar  nutation  is  also  given  by  Table 
XXVII. 


116  ASTRONOMY. 

283.  Tables  may  also  be  constructed  that  will  facilitate  the 
computation  of  the  radius  vector.     We  have, 

True  rad.  vector  =  elliptic  rad.  vector  +  perturbations. 

A  table  of  the  elliptic  radius  vector  may  be  formed  by  means  of 
the  polar  equation  of  the  orbit,  and  tables  of  the  perturbations 
from  their  analytical  expressions  (Art.  265).  The  tables  of  the 
perturbations  will  have  the  same  arguments  as  the  tables  of  the 
perturbations  of  longitude. 

284.  Lunar  and  planetary  tables  are  constructed  upon  the 
same  principles  as  the  solar  tables  we  have  been  describing, 
which  serve  to  make  known  the  orbit  longitude  and  radius 
vector.  But  other  tables  are  necessary  in  the  case  of  these 
bodies,  for  the  computation  of  the  ecliptic  longitude  and  the 
latitude. 

285.  The  difference  between  the  orbit  longitude  and  the  eclip- 
tic longitude,  is  called  the  Reduction  to  the  ecliptic.  A  formula 
for  the  reduction  has  been  investigated,  in  which  the  variable  is 
the  difference  between  the  orbit  longitude  and  the  longitude  of 
the  node.  If  this  formula  be  reduced  to  a  table,  by  taking  the 
reduction  from  the  table  and  adding  it  to  the  orbit  longitude  we 
shall  have  the  ecliptic  longitude.  Table  LIII  is  a  table  of  reduc- 
tion, for  the  moon. 

286.  For  the  latitude^  we  have  the  equation 

True  lat.  =  lat.  in  orbit  -f  perturbations. 
We  have  already  seen  (Art.  246),  that 
sin  (lat.  in  orbit)  =  sin  (orbit  long.  — -  long,  of  node),  sin  inclina. 

A  table  constructed  from  this  formula,  will  have  for  its  argument 
the  orbit  longitude  minus  the  longitude  of  the  node,  which  is  also 
the  argument  of  reduction.     (See  Table  LV). 

The  tables  of  the  perturbations  in  latitude  are  constructed  upon 
the  same  principles  as  the  tables  of  the  perturbations  in  longitude 
and  radius  vector. 

287.  A  table  exhibiting  the  longitude  and  latitude,  right  ascen- 
sion and  declination,  distance,  parallax,  semi-diameter,  &c.,  of  the 
sun  or  other  body,  at  stated  periods  of  time,  as  at  noon  of  each  day 
throughout  the  year,  is  called  an  Ephemeris  of  the  body.  An 
ephemeris  of  the  sun,  of  the  moon,  and  of  each  of  the  planets,  is 


MOTIONS    OP    COMETS    IN    THEIR    ORBITS.  117 

published  for  each  year  in  advance,  in  the  Enghsh  Nautical  Alma- 
nac, and  in  the  Connaissance  des  Tenis. 


CHAPTER    XI. 

OP    THE    MOTIONS    OF    THE    COMETS. 

288.  When  first  seen,  a  comet  is  ordinarily  at  some  distance 
from  the  sun,  and  moving  towards  him.  After  this  it  continues 
to  approach  the  sun  for  a  certain  time,  and  then  recedes  from 
him  to  a  greater  or  less  distance,  and  finally  disappears.  In 
many  instances  comets  have  come  so  near  the  sun  as  to  be  for  a 
time  lost  in  his  beams. 

289.  Comets  resemble  the  planets  in  their  changes  of  apparent 
place  amongst  the  fixed  stars,  but  they  differ  from  them  in  never 
having  been  observed  to  perform  an  entire  circuit  of  the  heavens. 
Their  apparent  motions  are  also  more  irregular  than  those  of  the 
planets,  and  they  are  confined  to  no  particular  region  of  the  hea- 
vens, but  traverse  indifferently  every  part. 

290.  Sir  Isaac  Newton,  from  observations  that  had  been  made 
upon  the  remarkable  comet  of  1680,  ascertained  that  this  comet 
described  a  parabolic  orbit,  having  the  sun  at  its  focus,  or  an  el- 
liptic orbit  of  so  great  an  eccentricity  as  to  be  undistinguishable 
from  a  parabola,  and  that  its  radius  vector  described  equal  areas 
in  equal  times.  Since  then,  the  orbits  of  about  140  comets  have 
been  computed,  and  found  to  be,  with  a  few  exceptions,  of  a 
parabolic  form,  or  sensibly  so. 

291.  It  was  demonstrated  by  Newton,  on  the  theory  of  gravi- 
tation, that  a  body  projected  in  space  may  describe  about  the 
sun  as  a  focus,  either  one  of  the  conic  sections,  and  that  the 
form  of  the  orbit  will  depend  upon  the  projectile  velocity  alone. 
With  one  particular  velocity  the  orbit  will  be  a  parabola  ;  with 
any  less  velocity,  it  will  be  an  ellipse  or  circle ;  and  with  any 


118  ASTRONOMY. 

greater  velocity,  it  will  be  a  hyperbola.  Now,  as  there  is  but  one 
velocity  from  which  a  parabolic  orbit  will  result,  and  as  any 
comet,  which  may  have  originally  moved  in  a  hyperbola,  must 
have  passed  its  perihelion,  and  receded  beyond  the  limits  of  the 
solar  system,  it  may  be  inferred,  with  great  probability,  that  the 
orbits  of  the  comets  whose  observed  courses  are  not  distinguish- 
able from  parabolic  arcs,  are  in  fact  ellipses  of  great  eccentri- 
city. This  is  the  theory  of  the  cometary  motions  proposed 
by  Newton. 

The  orbits  of  some  of  the  comets  are  known  from  observation 
to  be  very  eccentric  ellipses. 

292.  The  elements  of  a  comefs  orhit  are  the  longitude  of  the 
ascending  node,  the  inclination  of  the  orbit,  the  longitude  of  the 
perihelion,  the  perihelion  distance,  and  the  epoch  of  the  perihe- 
lion passage.  These  make  known  the  position  and  dimensions 
of  the  orbit,  on  the  supposition  that  it  is  a  parabola,  and  thus  ap- 
pertain only  to  the  motions  of  the  comet  for  the  period  during 
which  it  is  visible. 

293.  Assuming  that  the  radius  vector  of  a  comet  describes 
areas  proportional  to  the  times,  the  elements  of  its  orbit  may  be 
computed  from  three  observed  geocentric  places.  The  problem 
is,  however,  one  of  considerable  difficulty. 

294.  Astronomers  do  not  seek  to  deduce  from  the  observations 
made  during  one  appearance  of  a  comet,  its  entire  elliptic  orbit. 
It  is  impossible,  from  such  observations,  to  compute  the  major 
axis  of  its  orbit  and  its  period  with  any  accuracy,  inasmuch  as 
in  the  interval  during  which  they  are  made  the  comet  describes 
but  a  small  portion  of  its  entire  orbit. 

The  only  mode  of  obtaining  the  period  of  a  comet's  revolu- 
tion, is  by  noting  the  time  of  its  return  to  the  perihelion  of  its 
orbit.  A  comet  cannot  be  recognized  at  a  second  appearance  by 
its  aspect,  for,  this  is  liable  to  great  alterations.  It  may,  however, 
be  identified  by  means  of  the  elements  of  its  orbit,  as  it  is  ex- 
tremely improbable  that  the  elements  of  the  orbits  of  two  differ- 
ent comets  will  agree  throughout.  This  method  of  identifying 
a  comet  on  a  second  appearance  may  sometimes  fail  of  applica- 
tion, inasmuch  as  the  orbit  of  a  comet  may  experience  great  al- 
terations from  the  attractions  of  the  planets. 

295.  Owing  to  the  great  lengths  of  the  periods  of  most  of  the 


encke's  comet — biela's  comet — halley's  comet.     119 

comets,  and  the  comparatively  short  interval  during  which  their 
motions  have  been  carefully  observed,  there  are  but  three  comets, 
the  periods  and  entire  orbits  of  which  have  been  determined. 
These  are  denominated  Buckets  Co?net,  Biela's  Comet,  and  Hal- 
ley's  Comet.  The  two  former  are  small  telescopic  objects,  but  the 
latter,  when  near  its  perihelion,  is  distinctly  visible  to  the  naked 
eye. 

296.  Encke's  Comet  is  so  called  from  Professor  Encke,  of  Ber- 
lin, who  first  ascertained  its  periodical  return.  It  accomplishes 
its  revolution  in  the  short  period  of  1207  days,  or  about  3^  years, 
arid  moves  in  an  orbit  inclined  under  a  small  angle  to  the  plane  of 
the  ecliptic,  and  whose  perihelion  is  at  the  distance  of  the  planet 
Mercury,  and  aphelion  nearly  at  the  distance  of  Jupiter.  This 
discovery  was  made  on  the  occasion  of  its  fourth  recorded  appear- 
ance, in  1819.  Since  then,  it  has  returned  several  times  to  its 
perihelion,  and  in  every  instance  very  nearly  as  predicted.  Its 
last  return  took  place  in  1835,  its  next  will  happen  in  the  fall  of 
the  present  year  (1838). 

297.  Biela's  Comet,  as  it  is  called,  was  discovered  by  M.  Biela, 
of  Johannisberg,  on  the  27th  February,  1836.  Its  period  is  about 
6|  years.  Its  orbit  is  inclined  under  a  small  angle  to  the  plane  of 
the  ecliptic,  and  lies  mostly  between  the  orbits  of  the  earth  and  of 
Jupiter.  By  a  remarkable  coincidence,  the  orbit  of  this  comet 
very  nearly  intersects  the  orbit  of  the  earth.  Its  last  appearance 
took  place,  according  to  prediction,  in  1832 ;  the  next  will  be  in 
1838. 

298.  Halley's  Comet  is  so  called  from  Sir  Edmund  Halley, 
who  discovered  its  period,  and  correctly  predicted  its  return. 
From  a  comparison  of  the  elements  of  the  orbits  described  by  the 
comets  of  1531,  1607,  and  1682,  he  concluded  that  the  same 
comet  had  made  its  appearance  in  these  several  years,  and  pre- 
dicted that  it  would  again  return  to  its  perihelion  in  the  year  1759, 
as  it  actually  did.  Assuming  the  earth's  mean  distance  from  the 
sun  to  be  unity,  the  perihelion  distance  of  this  comet  is  0.58,  and 
aphelion  distance  35.32.  Accordingly,  it  approaches  the  sun  to 
within  one  half  the  distance  of  the  earth,  and  recedes  from  him 
far  beyond  the  orbit  of  Uranus.  Its  period  is  about  76  years,  but 
is  liable  to  a  variation  of  a  year  or  more,  from  the  effect  of  the  at- 
tractions of  the  planets.     The  last  perihelion  passage  took  place 


120  ASTRONOMY. 

on  the  16th  of  November,  1835,  within  a  few  days  of  the  pre- 
dicted time.     The  next  will  occur  about  the  year  1912. 

299.  Of  the  comets  which  have  been  observed,  some  have  a  di- 
rect and  others  a  retrograde  motion.  The  perihelia  of  their  or- 
bits, for  the  most  part,  lie  within  the  orbit  of  the  earth,  and  the 
aphelia  far  without  the  orbit  of  Uranus.  Many  of  them  come 
into  close  proximity  to  the  sun.  The  great  comet  of  1680,  ac- 
cording to  the  computation  of  Newton,  came  166  times  nearer 
the  sun  than  the  earth  is.  The  planes  of  the  orbits  are  inclined 
under  every  variety  of  angle  to  the  plane  of  the  ecliptic. 

300.  The  motions  of  the  comets  are  liable  to  great  derange- 
ments from  the  attractions  of  the  planets.  As  their  orbits  cross 
the  orbits  of  the  planets,  they  may  come  into  proximity  with 
these  bodies,  and  be  strongly  attracted  by  them.  The  comet  of 
1770,  commonly  called  Lexel's  Comet,  offers  a  striking  example 
of  the  disturbances  to  which  the  cometary  motions  are  exposed. 
From  observations  made  upon  this  comet  in  the  year  1770,  Lexel 
made  out  that  its  period  was  5^  years  ;  still  it  has  not  since  been 
seen.  According  to  Burckhardt,  this  comet,  previous  to  the  year 
1767,  moved  in  an  orbit  which  answered  to  a  period  of  50  years, 
and  never  approached  near  enough  to  the  earth  and  sun  to  become 
visible.  Early  in  the  year  1767  it  came  so  near  the  planet  Jupi- 
ter, that  his  attraction  changed  its  orbit  to  one  of  5^  years.  It 
thus  became  visible  in  1770,  and  would  have  again  been  seen  in 
1776,  had  it  not  been  so  situated  with  regard  to  the  earth  as  to  be 
entirely  hid  by  the  sun's  rays.  In  the  year  1779  it  again  met 
with  Jupiter,  and  its  orbit  was  so  much  enlarged  by  his  attraction, 
that  it  now  employs  twenty  years  in  completing  a  revolution,  and 
no  longer  comes  near  enough  to  the  earth  to  be  visible. 


MOTIONS    OF    THE    SATELLITES.  121 


CHAPTER    XII. 

OF    THE    MOTIONS    OF    THE    SATELLITES. 

301.  As  it  has  already  been  remarked,  the  planets  which  have 
satellites  are  Jupiter,  Saturn,  and  Uranus.  The  number  of  Ju- 
piter's satellites  is  four;  of  Saturn's,  seven  ;  of  Uranus',  six. 

302.  The  satellites  of  Jupiter  are  perceptible  with  telescopes 
of  moderate  power.  It  is  found  by  repeated  observations,  that 
they  are  continually  changing  their  positions  with  respect  to  one 
another  and  the  planet,  being  sometimes  all  to  the  right  of  the 
planet,  and  sometimes  all  to  the  left  of  it,  but  more  frequently 
some  on  each  side.  They  are  distinguished  from  each  other  by 
the  distance  to  which  they  recede  from  the  planet,  that  which 
recedes  to  the  least  distance  beings  called  the  F'irst  Satellite,  that 
which  recedes  to  the  next  greater  distance  the  Second,  and 
so  on. 

The  satellites  of  Jupiter  were  discovered  by  Galileo,  in  the 
year  1610. 

303.  The  satellites  of  Saturn  and  of  Uranus  cannot  be  seen 
except  through  excellent  telescopes.  They  experience  changes 
of  apparent  position,  similar  to  those  of  Jupiter's  satellites. 

304.  The  apparent  motion  of  Jupiter's  satellites  alternately 
from  one  side  to  the  other  of  the  planet,  leads  to  the  supposition 
that  they  actually  revolve  around  the  planet.  This  inference  is 
confirmed  by  other  phenomena.  While  a  satellite  is  passing 
from  the  eastern  to  the  western  side  of  the  planet,  a  small  dark 
spot  is  frequently  seen  crossing  the  disc  of  the  planet  in  the  same 
direction  :  and  again,  while  the  satellite  is  passing  from  the 
western  to  the  eastern  side,  it  often  disappears,  and  after  remaining 
for  a  time  invisible,  re-appears  at  another  place.  These  phe- 
nomena are  easily  explained,  if  we  suppose  that  the  planet  and 
its  satellites  are  opake  bodies,  illuminated  by  the  sun,  and  that 
the  satellites  revolve  around  the  planet  from  west  to  east.  On 
this  hypothesis,  the  dark  spot  seen  traversing  the  disc  of  the 

16 


122  ASTRONOMY. 

planet,  is  the  shadow  cast  upon  it  by  the  satellite  on  passing" 
between  the  planet  and  the  sun,  and  the  disappearance  of  the  sa- 
tellite is  an  eclipse^  occasioned  by  its  entering  the  shadow  of  the 
planet. 

As  the  transit  of  the  shadow  occurs  during  the  passage  of  the 
satellite  from  the  eastern  to  the  western  side  of  the  planet,  and  the 
eclipse  of  the  satellite  during  its  passage  from  the  western  to  the 
eastern  side,  the  direction  of  the  motion  must  be  from  west 
to  east. 

305.  Analogous  conclusions  may  be  drawn  from  similar  phe- 
nomena exhibited  by  the  satellites  of  Saturn.  The  satellites  of 
Uranus  also  revolve  around  their  primary,  but  the  direction  of 
their  motion  is  from  east  to  west. 

306.  Let  us  now  examine  into  the  principal  circumstances  of 
the  eclipses  of  Jupiter's  satellites,  and  of  the  transits  of  their 
shadows  across  the  disc  of  the  primary.  Let  E  E'  E"  (Fig.  47)  rep- 
resent the  orbit  of  the  earth,  P  P'  P"  the  orbit  of  Jupiter,  and  s  s'  s" 
that  of  one  of  its  satellites.  Suppose  that  E  is  the  position  of  the 
earth,  and  P  that  of  the  planet,  and  conceive  two  lines,  a  a',  b  6', 
to  be  drawn  tangent  to  the  sun  and  planet :  then,  while  the  satel- 
lite is  moving  from  s  to  s'  it  will  be  eclipsed,  and  while  it  is 
moving  from/ to/'  its  shadow  will  fall  upon  the  planet.  Again, 
if  E  c,  E  c'  represent  two  lines  drawn  from  the  earth  tangent  to 
the  planet  on  either  side,  the  satellite  will,  while  moving  from 
g  to  g',  traverse  the  disc  of  the  planet,  and  while  moving  from 
h  to  h'j  be  behind  the  planet,  and  thus  concealed  from  view. 
It  will  be  seen  on  an  inspection  of  the  figure,  that  during  the 
motion  of  the  earth  from  E",  the  position  of  opposition,  to  E', 
that  of  conjunction,  the  disappearances  or  iinmersions  of  the 
satellite  will  take  place  on  the  western  side  of  the  planet ;  and 
that  the  emersions,  if  visible  at  all,  can  be  so  only  when  the 
earth  is  so  far  from  opposition  and  conjunction  that  the  line  E  s'. 
drawn  from  the  earth  to  the  point  of  emersion,  will  lie  to  the  west 
of  E  c.  It  will  also  be  seen,  that  during  the  passage  of  the  earth 
from  E'  to  E"  the  emersions  will  take  place  on  the  eastern  side 
of  the  planet,  and  that  the  immersions  cannot  be  visible,  unless 
the  line  F  s,  drawn  from  the  earth  to  the  point  of  immersion, 
passes  to  the  east  of  the  planet.  It  appears  from  observation, 
that  the  immersion  and  emersion  are  never  both  visible  at  the 


PEBIODS    AND    MEAN    MOTIONS    OF    THE    SATELLITES.  123 

same   period,   except    in   the   case    of    the   third    and    fourth 
sateUites. 

If  the  orbits  of  the  satelUtes  lay  in  the  plane  of  Jupiter's  orbit, 
an  eclipse  of  each  satellite  would  occur  every  revolution,  but,  in 
point  of  fact,  they  are  somewhat  inclined  to  this  plane,  from 
which  cause  the  fourth  satellite  sometimes  escapes  an  eclipse. 

307.  The  periods  and  other  particulars  of  the  motions  of  the 
satellites,  result  from  observations  upon  their  eclipses.  The 
middle  point  of  time  between  the  satellite  entering  and  emerging 
from  the  shadow  of  the  primary,  is  the  time  when  the  satellite  is 
in  the  direction,  or  nearly  so,  of  a  line  joining  the  centres  of  the 
sun  and  primary.  If  the  latter  continued  stationary,  then  the 
interval  between  this  and  the  succeeding  central  eclipse  would 
be  the  periodic  time  of  the  satellite.  But,  the  primary  planet 
moving  in  its  orbit,  the  interval  between  two  successive  eclipses 
is  a  synodic  revolution.  The  synodic  revolution,  however, 
being  observed,  and  the  period  of  the  primary  being  known,  the 
periodic  time  of  the  satellite  may  be  computed. 

308.  The  mean  motions  of  the  satellites  differ  but  little  from 
their  true  motions  :  and  hence  the  forms  of  their  orbits  must  be 
nearly  circular.  The  orbit,  however,  of  the  third  satellite  of  Ju- 
piter has  a  small  eccentricity  ;  that  of  the  fourth,  a  larger. 

309.  The  distances  of  the  satellites  from  their  primary  are 
determined  from  micrometrical  measurements  of  their  apparent 
distances  at  the  times  of  their  greatest  elongations. 

A  comparison  of  the  mean  distances  of  Jupiter's  satellites  with 
their  periodic  times,  proves  that  Kepler's  third  law  with  respect  to 
the  planets  applies  also  to  these  bodies  ;  or,  that  the  squares  of 
their  sidereal  revolutions  are  as  the  cubes  of  their  mean  distances 
from  the  primary. 

The  same  law  also  has  place  with  the  satellites  of  Saturn  and 
Uranus. 

310.  The  computation  of  the  place  of  a  satellite  for  a  given 
time,  is  efiected  upon  similar  principles  with  that  of  the  place  of  a 
planet.  The  mutual  attractions  of  Jupiter's  satellites  occasion 
sensible  perturbations  of  their  motions,  of  which  account  must  be 
taken  when  it  is  desired  to  determine  their  places  with  accuracy. 

311.  Laplace  has  shown  from  the  theory  of  gravitation,  that,  by 
reason  of  the  mutual  attractions  of  the  first  three  of  Jupiter's  satel- 


iK^.'.M^ 


124  ASTRONOMY. 

lites,  their  mean  motions  and  mean  longitudes  are  permanently 
connected  by  the  following  reniarka])le  relations. 

1.  The  mean  motion  of  the  first  satellite  plus  twice  that  of  the 
third  is  equal  to  three  tiTnes  that  of  the  second. 

2.  The  mean  loiigitude  of  the  first  satellite  plus  ttvice  that  of 
the  third  minus  three  times  that  of  the  second  is  equal  to  180°. 

312.  It  follows  from  this  last  relation,  that  the  longitudes  of  the 
three  satellites  can  never  be  the  same  at  the  same  time,  and  conse- 
quently that  they  can  never  be  all  eclipsed  at  once. 


CHAPTER    XIII. 

ON    THE    MEASUREMENT    OF    TIME. 

Different  Kinds  of  Time. 

313.  In  Astronomy,  as  we  have  already  stated,  three  kinds  of 
time  are  used :  Sidereal,  True  or  Aj)parent  Solar,  and  Mean 
Solar  Time.  Sidereal  time  being  measured  by  the  diurnal  mo- 
tion of  the  vernal  equinox,  true  or  apparent  solar  time  by  that  of 
the  sun,  and  mean  solar  time  by  that  of  an  imaginary  sun  called 
the  Mean  sun,  conceived  to  move  uniformly  in  the  equator  with 
the  real  sun's  mean  motion  in  rio;ht  ascension  or  lonsfitude. 

314,  The  sidereal  day  and  the  mean  solar  day  are  each  of 
uniform  duration,  but  the  length  of  the  true  solar  day  is  varia- 
ble, as  we  will  now  proceed  to  show. 

The  sun's  daily  motion  in  right  ascension,  expressed  in  time, 
is  equal  to  the  excess  of  the  solar  over  the  sidereal  day.  Now 
this  arc,  and  therefore  the  true  solar  day,  varies  from  two 
causes,  viz : 

1.  The  inequality  of  the  sun's  daily  motion  in  longitude. 

2.  TJte  obliquity  of  the  ecliptic  to  the  equator. 

If  the  ecliptic  were  coincident  with  the  equator,  the  daily  arc 
of  right  ascension  would  be  equal  to  the  daily  arc  of  longitude, 


SOLAR    TIME — SIDEREAL    TIME.  125 

and  therefore  would  vary  between  the  limits  57'  11"  and  61'  10", 
which  would  answer  respectively  to  the  apogee  and  perigee. 
But,  owing  to  the  obliquity  of  the  ecliptic,  the  inclination  of  the 
daily  arc  of  longitude  to  the  equator  is  subject  to  a  variation ; 
and  this,  it  is  plain,  will  be  attended  with  a  variation  in  the 
daily  arc  of  right  ascension.  The  tendency  of  this  cause  is  ob- 
viously to  make  the  daily  arc  of  right  ascension  least  at  the 
equinoxes,  where  the  obliquity  of  the  arc  of  longitude  is 
greatest,  and  greatest  at  the  solstices,  where  the  obliquity  is 
least. 

315.  As  the  length  of  the  apparent  solar  day  is  variable,  it 
cannot  conveniently  be  employed  for  the  expression  of  intervals 
of  time  ;  moreover,  a  clock,  to  keep  apparent  solar  time,  requires 
to  be  frequently  adjusted.  These  inconveniences  attending  the 
use  of  apparent  solar  time,  led  astronomers  to  devise  a  new 
method  of  measuring  time,  to  which  they  gave  the  name  of 
mean  solar  time.  By  conceiving  an  imaginary  sun  to  move 
uniformly  in  the  equator  with  the  real  sun's  mean  motion,  a 
day  was  obtained,  of  which  the  length  is  invariable,  and  equal 
to  the  mean  length  of  all  the  apparent  solar  days  in  a  tropical 
year ;  and  by  supposing  the  right  ascension  of  this  fictitious 
sun  to  be,  at  the  instant  of  the  sun's  arrival  at  the  perigee  of  his 
orbit,  equal  to  the  sun's  true  longitude,  and  consequently  at  all 
times  equal  to  the  sun's  mean  longitude,  the  time  deduced  from 
its  position  with  respect  to  the  meridian,  was  made  to  corres- 
pond very  nearly  with  apparent  solar  time. 

316.  To  find  the  excess  of  the  mean  solar  day  over  the 
sidereal  day,  we  have  the  proportion 

360° :  24  sid.  hours  : :  59'  8".33  :  .r  =  3  m.  56.555  s. 
A  mean  solar  day,  comprising  24  mean  solar  hours,  is,  there- 
fore, 24h.  3m.  56.555s.  of  sidereal  time.     Hence,  a  clock  regu- 
lated to  sidereal  time  will  gain  3  m.  56.555  s.  in  a  mean  solar 
day. 

317.  In  order  to  find  the  expression  for  the  sidereal  day  in 
mean  solar  time,  we  must  use  the  proportion, 

24h.  3m.  56.555s. :  24h.  :  :  24h.  :  x  =  23h.  56m.  4.092s. 
The  dijfference  between  this  and  24  hours  is  3m.  55.908s.;  and 
therefore,  a  mean  solar  clock  will  lose  with  respect  to  a  sidereal 


120  ASTRONOMY. 

clock,  or  with  respect  to  the  fixed  stars,  3  m.  55.908  s.  m  a  si- 
dereal day,  and  proportionally  in  other  intervals.  This  is  called 
the  daily  acceleration  of  the  fixed  stars. 

318.  To  express  any  given  period  of  sidereal  time  in  mean 

solar  time,  we  must  subtract  for  each  hour  — —^ ■  =9.83«., 

and  for  minutes  and  seconds  in  the  same  proportion.     And,  on 

the  other  hand,  to  express  any  given  period  of  mean  solar  time 

3m  56  55s 
in  sidereal  time,  we  must  add  for  each  hour ^--^ — '=9.868., 

and  for  minutes  and  seconds  in  the  same  proportion. 

319.  It  is  the  practice  of  astronomers  to  adjust  the  sidereal 
clock  to  the  motions  of  the  true  instead  of  the  mean  equinox. 
The  inequality  of  the  diurnal  motion  of  this  point  is  too  small  to 
occasion  any  practical  inconvenience.  Sidereal  time,  as  deter- 
mined by  the  position  of  the  true  equinox,  will  not  deviate  from 
the  same  as  indicated  by  the  position  of  the  m.ean  equinox,  more 
than  2.3  s.  in  19  years. 

320.  Another  species  of  time,  called  Meati  Equinoctial  Tiine, 
has  recently  been  introduced  to  some  extent  into  astronomical 
calculations.  Mean  equinoctial  time  signifies  the  mean  time 
elapsed  since  the  instant  of  the  Mean  Vernal  Equinox.  Its  use 
is  to  afford  an  uniform  date,  which  shall  be  independent  of  the 
difiierent  meridians,  and  of  all  inequalities  in  the  sun's  motion, 
and  shall  thus  save  the  necessity,  when  speaking  of  the  time  of 
any  event's  happening,  of  mentioning  at  the  same  time  the  place 
where  it  was  observed  or  computed.  Thus,  it  is  the  same  thing 
to  say  that  a  comet  passed  its  perihelion  on  January  5th,  1837, 
at  5h.  47m.  0.0s.,  mean  time  at  Greenwich  ;  at  5h.  56m.  21.5s., 
mean  time  at  Paris ;  or  at  1836y.  289d.  6h.  16m.  40.96s.,  equi- 
noctial time  ;  buli^  the  former  dates  make  the  localities  of  Green- 
wich and  Paris  enter  as  elements  of  the  expression  ;  whereas  the 
latter  expresses  the  period  elapsed  since  an  epoch  common  to  all 
the  world,  and  identifiable  independently  of  all  localities.  By 
this  means  all  ambiguities  in  the  reckoning  of  time  are  supposed 
to  be  avoided. 

Conversion  of  One  Species  of  Time  into  Another. 

321.  The  difference  between  the  apparent  and  mean  time  is 
called  the  Equation  of  Time.     The  equation  of  time,  when 


CONVERSION    OF    APPARENT    INTO    MEAN    SOLAR    TIME.       127 

known,  serves  for  the  conversion  of  mean  time  into  apparent, 
and  the  reverse. 

322.  To  find  the  equation  of  time.  The  hour  angle  of  the 
sun  (p.  11,  def.  16)  varies  at  the  rate  of  360°  in  a  solar  day,  or 
15°  per  solar  hour.  If,  therefore,  its  value  at  any  moment  be  di- 
vided by  15,  the  quotient  will  be  the  apparent  time  at  that  mo- 
ment. In  like  manner,  the  hour  angle  of  the  mean  sun,  divided 
by  15,  gives  the  mean  time.  Now,  let  the  circle  V  S  D  (Fig. 
48)  represent  the  equator,  V  the  vernal  equinox,  M  the  point  of 
the  equator,  which  is  on  the  meridian,  and  V  S  the  right  ascen- 
sion of  the  sun,  and  we  shall  have, 

MS     VM— VS 

appar.  time  =  --—  = 

15  15 

Again,  if  we  suppose  the  circle  V  M  D  to  represent  the  mean 

equator,  V  the  mean  equinox,  and  S'  the  position  of  the  mean 

sun,  (V  S'  being  equal  to  the  mean  longitude  of  the  sun,)  we  shall 

have, 

MS'     VM  — VS'     VM— (V'S'  +  VV) 

mean  time  = = -— = -~^m • 

15  15  15 

thus, 

equa.  of  time=mean  time  —  ap.  time  = ^  \^^)'i 

JLO 

or,  the  equation  of  time  is  equal  to  the  difference  between  the 
siiii's  true  right  ascension  and  mean  longitude,  corrected  by 
the  equation  of  the  equinoxes  in  right  ascension,  and  con- 
verted into  time. 

A  formula  has  been  investigated,  and  reduced  to  a  table, 
which  makes  known  the  equation  of  time  by  means  of  the  sun's 
mean  longitude  as  an  argument.  (See  Table  XII.)  The  value 
of  the  equation  of  time  at  noon,  on  any  day  of  the  year,  is  also 
to  be  found  in  the  ephemeris  of  the  sun,  published  in  the  Nauti- 
cal Almanac,  and  other  works.  If  its  value  for  any  other  time 
than  noon  be  desired,  it  may  be  obtained  by  simple  proportion. 

The  value  of  the  equation  of  time,  determined  from  formula 
(74),  is  to  be  applied  with  its  sign  to  the  apparent  time  to  obtain 
the  mean,  and  with  the  opposite  sign  to  the  mean  time  to  obtain 
the  apparent.  

323.  The  equation  of  time  is  zero,  or  mean  and  true  time  are 
the  same  four  times  in  the  year,  viz  :  about  the  15th  of  April, 


128  ASTRONOMY. 

the  15th  of  June,  the  1st  of  September,  and  the  24th  of  Decem- 
ber. Its  greatest  additive  vahie  (to  apparent  time)  is  about  14^ 
minutes,  and  occurs  about  the  11th  of  February  ;  and  its  greatest 
subtractive  value  is  about  16\  minutes,  and  occurs  about  the  3d 
of  November. 

324.  To  convert  sidejeal  time  into  mean  time,  and  vice  versa. 
Making  use  of  Fig.  48  already  employed,  the  arc  V  M,  called  the 
Right  Ascension  of  Mid-Heaven,  expressed  in  time,  is  the  si- 
dereal time ;  V  S'  is  the  right  ascension  of  the  mean  sun,  esti- 
mated from  the  true  equinox,  or  the  mean  longitude  of  the  sun 
corrected  for  the  equation  of  the  equinoxes  in  right  ascension 
(Art.  322)  ;  and  M  S'  expressed  in  time,  is  the  mean  time.  Let 
the  arcs  V  M,  M  S'  and  V  S',  converted  into  time,  be  denoted 
respectively  by  S,  M,  and  L.     Now, 

V  M  =  M  S' -{- V  S' ; 
or,  S  =  M  +  L  .  .  (75) ;  and  M  =  S  —  L  .  .  (76). 

If  M  +  L  in  equation  (75)  exceeds  24  hours,  24  hours  must  be 
subtracted ;  and  if  L  exceeds  S  in  equation  (76),  24  hours  must 
be  added  to  S,  to  render  the  subtraction  possible. 

This  problem  may  in  practice  be  solved  most  easily  by  means  of 
an  ephemeris  of  the  sun,  which  gives  the  value  of  S,  or  the  side- 
real time  at  the  instant  of  mean  noon  of  each  day,  together  with  a 
table  of  the  acceleration  of  sidereal  on  mean  solar  time,  and  the 
corresponding  table  of  the  retardation  of  mean  on  sidereal  time. 

325.  The  conversion  of  apparent  time  into  sidereal,  or  sidereal 
time  into  apparent,  may  be  effected  by  first  obtaining  the  mean 
time,  and  then  converting  this  into  sidereal  or  apparent  time,  as 
the  case  may  be. 

Determination  of  the  Tim^e  and  Regulation  of  Clocks  by 
Astronomical  Observatioiis. 

326.  The  regulation  of  a  clock  consists  in  finding  its  ejTor  and 
its  rate. 

327.  The  error  of  a  mean  solar  clock  is  most  conveniently 
determined,  from  observations  with  a  transit  instrument  of  the 
time,  as  given  by  the  clock,  of  the  meridian  passage  of  the  sun's 
centre.  The  time  noted  will  be  the  clock  time  at  apparent  noon, 
and  the  exact  mean  time  at  apparent  noon  may  be  obtained  by 
applying  to  the  apparent  time  the  equation  of  time  with  its  proper 


DETERMINATION    OP    THE    TIME.  129 

sioTi,  which  may  for  this  purpose  be  taken  from  the  Nautical  Alma- 
nac by  simple  inspection,  A  comparison  of  the  clock  time  with 
the  exact  mean  time  will  give  the  error  of  the  clock. 

328.  The  daily  rate  of  a  mean  solar  clock  may  be  ascertained 
by  finding  as  above  the  error  at  two  successive  apparent  noons. 
If  the  two  errors  are  the  same  and  lie  the  same  way,  the  clock 
goes  accurately  to  mean  solar  time  ;  if  they  are  different,  their 
difference  or  sum,  according  as  they  lie  the  same  or  opposite 
ways,  will  be  the  daily  gain  or  loss,  as  the  case  may  be. 

329.  To  find  the  error  of  a  sidereal  clock,  compute  the  true 
right  ascension  of  some  one  of  the  fixed  stars,  (see  Prob.  XXI,) 
and  note  the  time  of  its  transit ;  the  difference  between  the  time 
observed  and  the  right  ascension  in  time  will  be  the  error.  The 
error  of  the  daily  rate  is  determined  by  observing  two  successive 
transits  of  the  same  star.  The  variation  of  the  time  of  the  second 
transit  from  that  of  the  first  will  be  the  error  in  question. 

The  error  and  rate  may  be  determined  more  accurately  from 
observations  upon  several  stars,  taking  a  mean  of  the  individual 
results.  Stars  at  a  distance  from  the  pole  are  to  be  selected,  for 
reasons  which  have  been  already  assigned. 

330.  In  default  of  a  transit  instrument,  the  time  may  be 
obtained,  and  time-keepers  regulated,  by  observations  made  out 
of  the  meridian.  There  are  two  methods  by  which  this  may 
be  accomplished,  called,  respectively,  the  method  of  Single 
Altitudes,  and  the  method  of  Double  Altitudes  or  of  Equal 
Altitudes.     These  we  will  now  explain. 

1.  To  determine  the  time  from  a  measured  altitude  of  the 
sun,  or  of  a  star,  its  declifiation  and  also  the  latitude  of  the 
place  being  given. 

Let  us  first  suppose  that  the  altitude  of  the  sun  is  taken  ; 
correct  the  measured  altitude  for  refraction  and  parallax,  and 
also,  if  the  sextant  is  the  instrument  used,  for  the  semi- 
diameter  of  the  sun.  Then,  if  Z  (Fig.  13)  represents  the 
zenith,  P  the  elevated  pole,  and  S  the  sun  ;  in  the  triangle  Z  P  S 
we  shall  know  Z  P  =  co-latitude,  P  S  =  co-declination,  and  Z  S  = 
co-altitude,  from  which  we  may  compute  the  angle  Z  P  S  (=  P)j 
which  is  the  angular  distance  of  the  sun  from  the  meridian, 
or,  if  expressed  in  time,  the  time  of  the  observation  from  appa- 
rent noon,  by  the  following  equations  (App.,  Resolution  of 
oblique-angled  spherical  triangles,  Case  I), 
17 


130  ASTRONOMY. 

2yt  =  ZP+PS  +  ZS  =  co-lat.  +  co-dec.  +  co-alt.  .  .  .  (77). 

sill  Z  P  sin  P  S 

or        siii^  ip  =  sin  (A:  —  co-lat.)  sin  {k  —  co-dec.)       _  _  ,^g^ 
'  ^  sin  (co-lat.)  sin  (co-dec.) 

The  value  of  P  being  derived  from  these  equations  and  con- 
verted into  time  (see  Prob.  Ill),  the  result  will  be  the  appa- 
rent time  at  the  instant  of  the  observation,  if  it  was  made  in  the 
afternoon ;  if  not,  what  remains  after  subtracting  it  from  24 
hours  will  be  the  apparent  time.  The  apparent  time  being 
found,  the  mean  time  may  be  deduced  from  it  by  applying  the 
equation  of  time. 

A  more  accurate  result  will  be  obtained  if  several  altitudes 
be  measured,  the  time  of  each  measurement  noted,  and  the  mean 
of  all  the  altitudes  taken  and  regarded  as  corresponding  to  the 
mean  of  the  times.  The  correspondence  will  be  sufficiently 
exact  if  the  measurements  be  all  made  within  the  space  of  10 
or  12  minutes,  and  when  the  sun  is  near  the  prime  vertical. 
If  an  even  number  of  altitudes  be  taken,  and  alternately  of  the 
upper  and  lower  limb,  the  mean  of  the  whole  will  give  the  alti- 
tude of  the  sun's  centre,  without  it  being  necessary  to  know 
his  apparent  semi-diameter.  In  practice,  the  declination  of  the 
sun  may  be  taken  for  the  solution  of  this  problem  from  an 
ephemeris  of  the  sun.  For  this  purpose  the  time  of  the  obser- 
vation and  the  longitude  of  the  place  must  be  approximately 
known. 

Example.  On  the  1st  of  June,  1838,  at  about  10  h.  45  m. 
A.  M.,  the  altitude  of  the  sun's  lower  limb  was  measured  at 
New  York  with  a  sextant,  and  found  to  be  64°  55'  5".  What 
was  the  correct  time  of  the  observation  ? 

Measured  alt.  of  sun's  lower  limb,     .  64°  55'     5" 

Sun's  semi-diam.,  by  Conn,  des  Tems,  15   48 

Appar.  alt.  of  sun's  centre, 
Parallax  in  alt.  (Table  X), 
Refraction  (Table  VIII), 

True  alt.  of  sun's  centre, 


65 

10 

53 

+  4 

- 

-27 

65 

10 

30 

DETERMINATION    OF    TIME    BY    SINGLE    ALTITUDES.       131 

N.  York  approx.  time  of  observation,     10  h.  45  m. 
Diff.  of  long,  of  Paris  and  N.  York,         5         5 


Paris,  approx.  time  of  obs.      .         .         4       20      P.  M. 

Sun's  declin.  June  1st,  M.  noon  at  Paris,     22°     2'  27" 
"         '•       June  2d,  "  "  22    10    31 


Change  of  declin.  in  24  hours,  . 

'  24h.  :  8'  4"  : :  4h.  20m.  :  1'  27". 

Declin.  June  1st,  M,  noon  at  Paris,     . 
Change  of  declin.  in  4h.  20  m. 


8     4 


22°    2'  27" 
-fl    27 


Declin.  at  time  of  obs. 

90°     0'    0" 
Lat.  of  N.  York,  40    42   40 


Co-lat.      .      .    49  17  20 

Co-dec.    .      .     67  56  6 

Co-alt.      .       .     24  49  30 

2)142  2  56 


22      3   54 


logarithms, 
ar.  CO.  sin.  0.12033 
ar.  CO.  sin.  0.03303 


k 

. 

.  71 

1  28 

k- 

-  co-lat. 

.  21 

44  8  .  . 

.  sin.  9.56858 

k- 

-  co-dec. 

.   3 

5  22  .  . 

.  sin.  8.73154 

^P=  9  42  13  . 

P  =  19  24  26 
4 

Ih.  17m.  37s.  44'" 

10  42  22  A.M. 
Equa.  of  time,    —  2  34 


2  )  18.45348 
sm.  9.22674 


M.  time  of  obs.  10  39  48  A.M. 


132  ASTRONOMY. 

In  case  the  altitude  of  a  star  is  taken,  the  value  of  P  derived 
from  formula  (79),  when  converted  into  time,  will  express  the 
distance  in  time  of  the  star  from  the  meridian,  and  being  added 
to  the  right  ascension  of  the  star,  if  the  observation  be  made  to 
the  westward  of  the  meridian,  or  subtracted  from  the  right 
ascension  (increased  by  24h.,  if  necessary,)  if  the  observation 
be  made  to  the  eastward,  will  give  the  sidereal  time  of  the  ob- 
servation. 

2.  To  determine  the  time  of  noon  from  equal  altitudes  of  the 
sun,  the  times  of  the  observations  being  given. 

If  the  sun's  declination  did  not  change  while  he  is  above  the 
horizon,  he  would  have  equal  altitudes  at  equal  times  before 
and  after  apparent  noon.  Hence,  if  to  the  time  of  the  first 
observation  one  half  the  interval  of  time  between  the  two  obser- 
vations should  be  added,  the  result  would  be  the  time  of  noon, 
as  shown  by  the  clock  or  watch  employed  to  note  the  times  of 
the  observations.  The  deviation  from  12  o'clock  would  be  the 
error  of  the  clock  with  respect  to  apparent  time.  The  difference 
between  this  error  and  the  equation  of  time  would  be  the  error 
of  the  clock  with  respect  to  mean  time. 

But,  as  in  point  of  fact  the  sun's  declination  is  continually 
changing,  equal  altitudes  will  not  have  place  precisely  at  equal 
times  before  and  after  noon,  and  it  is  therefore  necessary  in  order 
to  obtain  an  exact  result,  to  apply  a  correction  to  the  time  thus 
obtained.  This  correction  is  called  the  Equation  of  Equal 
Altitudes.  Tables  have  been  constructed,  by  the  aid  of  which 
the  equation  is  easily  obtained.  This  is  at  the  same  time  a  very 
simple  and  very  accurate  method  of  finding  the  time,  and  the 
error  of  a  clock. 

If  equal  altitudes  of  a  star  should  be  observed,  it  is  evident 
that  half  the  interval  of  time  elapsed  would  give  the  time  of 
the  star  passing  the  meridian,  without  any  correction.  From 
this  the  error  of  the  clock  (if  keeping  sidereal  time)  may  be 
found,  as  explained  in  Art.  329. 

Of  the  Calendar. 
331.  The  sun  naturally  regidates  the  beginnings,  ends,  and 
durations  of  the  seasons  ;  and  the  calendar  is  constructed  to  dis- 
tribute and  arrange  the  smaller  portions  of  the  year. 


CALENDAR.  133 

332.  The  calendar  divides  the  year  into  12  months,  contain- 
ing in  all  365  days.  Now,  it  is  desirable  that  it  should  always 
denote  the  same  parts  of  the  same  season  by  the  same  days  of 
the  same  months,  that,  for  instance,  the  summer  and  winter  sol- 
stices, if  once  happening  on  the  21st  of  June  and  21st  of  Decem- 
ber, should  ever  after  be  reckoned  to  happen  on  the  same  days  ; 
that  the  date  of  the  sun's  entering  the  equinox,  the  natural  com- 
mencement of  spring,  should,  if  once,  be  always  on  the  20th  of 
March.  For  thus  the  labours  of  agriculture,  which  really  de- 
pend on  the  situation  of  the  sun  in  the  heavens,  would  be  simply 
and  truly  regulated  by  the  calendar. 

This  would  happen,  if  the  civil  year  of  365  days  were  equal 
to  the  astronomical  ;  but  the  latter  is  greater  ;  therefore,  if  the 
calendar  should  invariably  distribute  the  year  into  365  days,  it 
would  fall  into  this  kind  of  confusion,  that  in  progress  of  time, 
and  successively,  the  vernal  equinox  would  happen  on  every 
day  of  the  civil  year.     Let  us  examine  this  more  nearly. 

Suppose  the  excess  of  the  astronomical  year  above  the  civil  to 
be  exactly  6  hours,  and,  on  the  noon  of  March  20th  of  a  certain 
year,  the  sun  to  be  in  the  equinoctial  point ;  then,  after  the  lapse 
of  a  civil  year  of  365  days,  the  sun  would  be  on  the  meridian, 
but  not  in  the  equinoctial  point ;  it  would  be  to  the  west  of  that 
point,  and  would  have  to  move  6  hours  in  order  to  reach  it,  and 
to  complete  the  astronomical  or  tropical  year.  At  the  comple- 
tions of  a  second  and  a  third  civil  year,  the  sun  would  be  still 
more  and  more  remote  from  the  equinoctial  point,  and  would  be 
obliged  to  move,  respectively,  for  12  and  18  hours  before  he 
could  rejoin  it  and  complete  the  astronomical  year. 

At  the  completion  of  a  fourth  civil  year  the  sun  would  be  more 
distant  than  on  the  two  preceding  ones,  from  the  equinoctial 
point.  In  order  to  rejoin  it,  and  to  complete  the  astronomical 
year,  he  must  move  for  24  hours  ;  that  is,  for  one  whole  day.  In 
other  words,  the  astronomical  year  would  not  be  completed  till 
the  beginning  of  the  next  astronomical  day  ;  till,  in  civil  reckon- 
mg,  the  noon  of  March  2\st. 

At  the  end  of  four  more  common  civil  years,  the  sun  would 
be  in  the  equinox  on  the  noon  of  March  22d.  At  the  end  of  8 
and  64  years,  on  March  23d  and  April  6th,  respectively ;  at  the 
end  of  736  years,  the  sun  would  be  in  the  vernal  equinox  on 


134  ASTRONOMY. 

September  20th  ;  and  in  a  period  of  about  1508  years,  the  sun 
would  have  been  in  every  sign  of  the  zodiac  on  the  same  day  of 
the  calendar,  and  in  the  same  sign  on  every  day, 

333.  If  the  excess  of  the  astronomical  above  the  civil  year 
were  really  what  we  have  supposed  it  to  be,  6  hours,  this  confu- 
sion of  the  calendar  might  be  most  easily  avoided.  It  would  be 
necessary  merely  to  make  every  fourth  civil  year  to  consist  of 
366  days  ;  and,  for  that  purpose,  to  interpose,  or  to  intercalate,  a 
day  in  a  month  previous  to  March.  By  this  intercalation^ 
what  would  have  been  March  21st  is  called  March  20th,  and 
accordingly  the  sun  would  be  still  in  the  equinox  on  the  same 
day  of  the  month. 

334.  This  mode  of  correcting  the  calendar  was  adopted  by 
Julius  Caesar.  The  fourth  year  into  which  tlie  intercalary  day 
is  introduced  was  called  Bissextile  ;  it  is  now  frequently  called 
the  Leap  year.  The  correction  is  called  the  Julian  correctioUj 
and  the  length  of  a  mean  Julian  year  is  365 d.  6h. 

By  the  Julian  Calendar,  every  year  that  is  divisible  by  A:  is  a 
leap  year,  and  the  rest  com/mmi  years. 

335.  The  astronomical  year  being  equal  to  365  d.  5h.  48  m. 
47.6  s.,  it  is  less  than  the  mean  Julian  by  11m.  12.4s.  or 
0.007783d.  The  Julian  correction,  therefore,  itself  needs  correc- 
tion. The  calendar  regulated  by  it,  would,  in  process  of  time, 
become  erroneous,  and  would  require  reforrnation. 

The  intercalation  of  the  Julian  correction  being  too  great,  its 
effect  would  be  to  antedate  the  happening  of  the  equinox. 
Thus  (to  return  to  the  old  illustration)  the  sun,  at  the  comple- 
tion of  the  fourth  civil  year,  now  the  Bissextile,  would  have 
passed  the  equinoctial  point  by  a  time  equal  to  four  times 
0.007783d. ;  at  the  end  of  the  next  Bissextile,  by  eight  times 
0.007783  d.  ;  at  the  end  of  130  years,  by  about  one  day.  In 
other  words,  the  sun  would  have  been  in  the  equinoctial  point 
24  hours  previously,  or  on  the  noon  of  March  19^A. 

In  the  lapse  of  ages  this  error  would  continue  and  be  in- 
creased. Its  accumulation  in  1300  years  would  amount  to  10 
days,  and  then  the  vernal  equinox  would  be  reckoned  to  happen 
on  March  10th. 

336.  The  error  into  which  the  calendar  had  fallen,  and  would 
continue  to  fall,  was  noticed  by  Pope  Gregory  in  1582.     At  his 


CALENDAR.  135 

time  the  length  of  the  year  was  known  to  greater  precision  than 
at  the  time  of  Julius  Csesar.  It  was  supposed  equal  to  365 d. 
5h.  49  m.  16.23  s.  Gregory,  desirous  that  the  vernal  equinox 
should  be  reckoned  on  or  near  March  21st  (on  which  day  it 
happened  in  the  year  325,  when  the  Council  of  Nice  was  held), 
ordered  that  the  day  succeeding  the  4th  of  October,  1582,  in- 
stead of  being  called  the  5th,  should  be  called  the  15th  ;  thus 
suppressing  10  days,  which,  in  the  interval  between  the  years 
325  and  1582,  represented  nearly  the  accumulation  of  error 
arising  from  the  excessive  intercalation  of  the  Julian  correction. 
This  act  reformed  the  calendar.  In  order  to  correct  it  in  fu- 
ture ages,  it  was  prescribed  that,  at  certain  convenient  periods, 
the  intercalary  day  of  the  Julian  correction  should  be  omitted. 
Thus  the  centurial  years  1700,  1800,  1900,  are,  according  to  the 
Julian  calendar,  Bissextiles,  but  on  these  it  was  ordered  that  the 
intercalary  day  should  not  be  inserted ;  inserted  again  in  2000, 
but  not  inserted  in  2100,  2200,  2300  ;  and  so  on  for  succeeding 
centuries.  Bi/  the  Gregorian  calendar,  then,  everi/  centurial 
year  that  is  divisible  by  400  is  a  Bissextile  or  Leap  year,  and 
the  otliers  common  years.  For  other  than  centurial  years,  the 
rule  is  the  same  as  with  the  Julian  calendar. 

337.  This  is  a  most  simple  mode  of  regulating  the  calendar. 
It  corrects  the  insufficiency  of  the  Julian  correction,  by  omitting, 
in  the  space  of  400  years,  3  intercalary  days.  And  it  is  easy  to 
estimate  the  degree  of  its  accuracy.  For,  the  real  error  of  the 
Julian  correction  is  0.007783  d.  in  1  year,  consequently  400  x 
0.007783d.  or  3.1132d.  in  400  years.  Consequently,  0.1132d.  or 
2h.  43m.  0.5s.  in  400  years,  or  1  day  in  3533  years,  is  the  mea- 
sure of  the  degree  of  inaccuracy  in  the  Gregorian  correction. 

338.  The  Gregorian  calendar  was  adopted  immediately  on  its 
promulgation,  in  all  Catholic  countries,  but  in  those  where  the 
Protestant  religion  prevailed,  it  did  not  obtain  a  place  till  some 
time  after.  In  England,  "  the  change  of  style,"  as  it  was  called, 
took  place  after  the  2d  of  September,  1752,  eleven  nominal  days 
being  then  struck  out ;  so  that  the  last  day  of  Old  Style  being  the 
2d,  the  first  oi  New  Style  (the  next  day)  was  called  the  14th,  in- 
stead of  the  3d.  The  same  legislative  enactment  which  estab- 
lished the  Gregorian  calendar  in  England,  changed  the  time  of 
the  beginning  of  the  year  from  the  25th  of  March  to  the  1st  of 


136  ASTRONOMY. 

January.  Thus  the  year  1752,  which  by  the  old  reckoning 
would  have  commenced  with  the  25th  of  March,  was  made  to 
begfin  with  the  1st  of  January :  so  that  the  number  of  the  year 
is,  for  dates  falling  between  the  1st  of  January  and  the  25th  of 
March,  one  greater  by  the  new  than  by  the  old  style.  In  con- 
sequence of  the  intercalary  day  omitted  in  the  year  1800,  there 
is  now,  for  all  dates,  12  days  difference  between  the  old  and 
new  style. 

Russia  is  at  present  the  only  Christian  country  in  which  the 
Gregorian  calendar  is  not  used. 

339.  The  calendar  months  consist,  each  of  them,  of  30  or  31 
days,  except  the  second  month,  February,  which,  in  a  common 
year,  contains  28  days,  and  in  a  Bissextile,  29  days  ;  the  inter- 
calary day  being  added  at  the  last  of  this  month. 

340.  To  find  the  number  of  days  comprised  in  any  number 
of  civil  years,  multiply  365  by  the  number  of  years,  and  add  to 
the  product  as  many  days  as  there  are  Bissextile  years  in  the 
period. 


PART  11. 


ON  THE  PHENOMENA  RESULTING  FROM  THE  MO- 
TIONS OF  THE  HEAVENLY  BODIES,  AND  ON 
THEIR    APPEARANCES,    DIMENSIONS, 
AND  PHYSICAL  CONSTITUTION. 


CHAPTER    XIV. 

OF    THE    SUN    AND    THE    PHENOMENA     ATTENDING     ITS    APPA- 
RENT   MOTIONS. 

Inequality  of  Days* 
341.  At  all  places  north  or  south  of  the  equator  the  observer  is 
in  an  oblique  sphere ;  the  celestial  equator,  and  the  parallels  of 
declination  are  oblique  to  the  horizon  ;  and  the  equator  is  bisect- 
ed by  the  horizon,  while  the  parallels  are  divided  by  it  into  une- 
qual parts.  This  position  of  the  sphere  is  represented  in  Fig.  7, 
where  H  O  R  is  the  horizon,  Q.  O  E  the  equator,  and  n  c  r,  s  c  t 
(fcc.  parallels  of  declination.  Now,  the  length  of  the  day  is  mea- 
sured by  the  portion  of  the  parallel  to  the  equator  described  by  the 
sun,  which  lies  above  the  horizon.  On  inspecting  Fig-.  7,  it  will 
be  evident  that  this  diixdnishes  continually  while  the  sun  is  mov- 
ing from  the  position  of  greatest  northern  declination  to  that  of 
greatest  southern  declination,  and  increases  continually  while  he 
is  moving  from  the  position  of  greatest  southern  to  that  of  great- 
est northern  declination.  Whence  it  appears  that  the  day  will 
diminish  in  length  from  the  summer  to  the  winter  solstice,  and 
increase  in  length  from  the  winter  to  the  summer  solstice. 

•  The  day,  here  considered,  is  the  interval  between  sunrise  and  sunset. 

18 


138  ASTRONOMY, 

342.  As  the  equator  is  bisected  by  the  horizon,  at  the  equinoxes 
the  day  and  night  must  be  each  12  hours  long. 

343.  When  the  sun  is  north  of  the  equator,  the  greater  part  of 
its  diurnal  circle  lies  above  the  horizon,  in  northern  latitudes  ; 
and  therefore  from  the  vernal  to  the  autumnal  equinox  the  day 
is,  in  the  northern  hemisphere,  more  than  12  hours  in  length. 
On  the  other  hand,  when  the  sun  is  south  of  the  equator,  the 
greater  part  of  its  circle  lies  below  the  horizon,  and  hence  from 
the  autumnal  to  the  vernal  equinox  the  day  is  less  than  12 
hours  in  length. 

In  the  latter  interval  the  nights  will  obviously,  at  correspond- 
ing periods,  be  of  the  same  length  as  the  days  in  the  former. 

344.  The  variation  in  the  length  of  the  day  in  the  course  of 
the  year,  will  increase  with  the  latitude  of  the  place  ;  for  the 
greater  is  the  latitude  the  more  oblique  are  the  circles  described 
by  the  sun  to  the  horizon,  and  the  greater  is  the  disparity  be- 
tween the  parts  into  which  they  are  divided  by  the  horizon. 
This  will  be  obvious,  on  referring  to  Fig.  7,  where  H  O  R, 
H'  O  R',  represent  the  positions  of  the  horizons  of  two  different 
places  with  respect  to  these  circles,  H'  O  R'  being  the  hori- 
zon for  which  the  latitude  or  the  altitude  of  the  pole  is  the 
greatest. 

For  the  same  reason,  the  days  will  be  the  longer  as  we  proceed 
from  the  equator  northward,  during  the  period  that  the  sun  is 
north  of  the  equinoctial,  and  the  shorter,  during  the  period 
that  he  is  south  of  this  circle. 

345.  At  the  equator  the  horizon  bisects  all  the  diurnal  circles, 
and  consequently  the  day  and  night  are  there  each  12  hours  in 
length  throughout  the  year. 

346.  At  the  arctic  circle  the  day  will  be  24  hours  long  at  the 
time  of  the  summer  solstice  :  for,  the  polar  distance  of  the  sun 
will  then  be  66^°,  which  is  the  same  as  the  latitude  of  the 
arctic  circle,  whence  it  follows  that  the  diurnal  circle  of  the  sun 
at  this  epoch,  will  correspond  to  the  circle  of  perpetual  appari- 
tion for  the  parallel  in  question. 

On  the  other  hand,  when  the  sun  is  at  the  winter  solstice,  the 
night  will  be  24  hours  long  on  the  arctic  circle. 

347.  To  the  north  of  the  arctic  circle,  the  sun  will  remain 
continually  above  the  horizon,  during  the  period  that  his  north 


DETERMINATION    OF    THE    LENGTH    OF    THE    DAY.  139 

polar  distance  is  less  than  the  latitude  of  the  place,  and  continu- 
ally  below  the  horizon  during  the  period  that  his  south  polar  dis- 
tance is  less  than  the  latitude  of  the  place. 

At  the  north  pole,  as  the  horizon  is  coincident  with  the  equa- 
tor, the  sun  will  be  above  the  horizon  while  passing  from  the  ver- 
nal to  the  autumnal  equinox,  and  below  it  while  passing  from 
the  autumnal  to  the  vernal  equinox.  Accordingly,  at  this  locality 
there  will  be  but  one  day  and  one  night  in  the  course  of  a  year, 
and  each  will  be  of  six  months'  duration. 

348.  The  circumstances  of  the  duration  of  light  and  dark- 
ness are  obviously  the  same  in  the  southern  hemisphere  as  in 
the  northern,  for  corresponding  latitudes  and  corresponding  de- 
clinations of  the  sun. 

349.  The  latitude  of  the  iilace  and  the  declination  of  the  sun 
being  given,  to  find  the  times  of  the  smi's  rising  and  setting 
and  the  length  of  the  day. 

Let  H  P  R  (Fig.  49)  be  the  meridian,  H  M  R  the  horizon,  and 
B  5  D  the  diurnal  circle  described  by  the  sun.  The  hour  angle 
E  P  ^,  or  its  measure  E  t,  which  converted  into  time  expresses 
the  interval  between  the  rising  or  setting  of  the  sun  and  his  pas- 
sage over  the  meridian,  is  called  the  Semi-diurnal  Arc.     Now, 

E;  =  EM-+-M#  =  90°  +  M^; 

which  gives,  cos  E  ^  =  —  sin  M  ^ ;  /  /y6^"^ 

and  we  have  by  Napier's  rules, 

sin  M  ^  =  cot  ^  M  5  tang  t  s  =  tang  P  M  H  tang  E  B  =  tang  P  H 

tang  E  B ; 

whence,  cos  E  ^  =  —  tang  P  H  tang  E  B, 

or, 

cos  (semi-diurnal  arc)  =  —  tang  lat.  x  tang  dec.  ,  .  .  (80). 

The  semi-diurnal  arc  (in  time)  expresses  the  apparent  time  of 
the  sun's  setting ;  and  subtracted  from  12  hours,  gives  the  appa- 
rent time  of  its  rising.  The  double  of  it  will  be  the  length  of 
the  day. 

In  resolving  this  problem  it  will,  in  practice,  generally  answer 
to  make  use  of  the  declination  of  the  sun  at  noon  of  the  given 
day,  which  may  be  taken  from  an  ephemeris. 

Exam.  1.  Let  it  be  required  to  find  the  apparent  times  of  the 


4 


^, 


i    f  ■    / 
140  ASTRONOMY. 

sun's  rising  and  setting,  and  the  length  of  the  day  at  New  York, 

at  the  summer  solstice. 

Log.  tang  lat.  (40°  42'  40")       .         .        9.93474  — 
Log.  tang  dec.  (23°  27'  40")      .         .         9.63749 


Log.  cos  (semi-diurnal  arc)        .         .         9.57223  — 
Semi-diurnal  arc       .         .         .         .111°   55'     40" 
Time  of  sun's  setting         .         .         .         7h.  27m.  43s. 
Time  of  sun's  rising  .         .         .         4     32      17 

Length  of  day  .         .         .         .       14     55      26 

Exam.  2.  What  are  the  lengths  of  the  longest  and  shortest 
days  at  Boston ;  the  latitude  of  that  place  being  42°  21'  15"  N. 
Ans.  15h.  6m.  28s.  and  8h.  53m.  32s. 

Exam.  3.  At  what  hours  did  the  sun  rise  and  set  on  May  1st, 
1837,  at  Charleston  ;  the  latitude  of  Charleston  being  32°  47', 
and  the  declination  of  the  sun  being  15°  6'  0"  N. 

Ans.  Time  of  rising,  5h.  19m.  58s.    Time  of  setting,  6h.40m.2s. 
350.    To  find  the  time  of  the  sun's  apparent  rising  or  setting  : 
the  latitude  of  the  place  and  the  declination  of  the  sun  being 
given. 

At  the  time  of  his  apparent  rising  or  setting,  the  sun  as  seen 
from  the  centre  of  the  earth  will  be  below  the  horizon  a  dis- 
tance s  S  (Fig.  49)  equal  to  the  refraction  minus  the  parallax. 
The  mean  difference  of  these  quantities  is  33'  42'^  Let  it  be 
denoted  by  R.  Now  to  find  the  hour  angle  Z  P  S  (=  P),  the 
triangle  Z  P  S  gives  (see  Appendix), 

^      Z  P  +  PS  +Z  S_co.lat.+co.dec.  +  (90°+R)         ,^.. 

sin^  i  P  =  sin  {k  —  Z  P)  sin  {k  —  PS) . 
-  sin  Z  P  sin  P  S  ' 

or,  .in^  1  P-  sin  {k  -  CO.  lat.).  sin  {k  ~  co.  dec.)   ^  ^ 

sin  (co.  lat.).  sin  (co.  dec) 

The  value  of  P,  (in  time),  will  be  the  interval  between  appa- 
rent noon  and  the  time  of  the  apparent  risincr  or  settino-. 

If  the  time  of  the  rising  or  setting  of  the  upper  limb  of  the  sun, 
instead  of  its  centre,  be  required,  we  must  take  for  R,  33'  42"  -\- 
sun's  semi-diameter,  or  49'  43". 

Unless  very  accurate  results  are  desired,  it  will  be  sufficient  to 


TWILIGHT.  141 

take  the  declinations  of  the  sun  at  6  o'clock  in  the  morning  and 
evening.  When  the  greatest  precision  is  required,  the  times  of 
true  rising  and  setting  must  be  computed  by  equation  (80),  and 
the  declinations  found  for  these  times. 

Twilight. 

351.  When  the  sun  has  descended  below  the  horizon,  its  rays 
still  continue  to  fall  upon  a  certain  portion  of  the  body  of  air 
that  lies  above  it,  and  are  thence  reflected  down  upon  the  earth, 
so  as  to  occasion  a  certain  degree  of  light,  which  gradually 
diminishes,  as  the  sun  descends  farther  below  the  horizon  and 
the  portion  of  the  air  posited  above  the  horizon,  that  is  directly 
illuminated,  becomes  less.  The  same  effect,  though  in  a  reverse 
order,  takes  place  in  the  morning  previous  to  the  sun's  rising. 
The  light  thus  produced  is  called  the  Crepiisculum^  or 
Twilight. 

352.  The  close  of  the  evening  twilight  is  marked  by  the 
appearance  of  faint  stars  over  the  western  horizon,  and  the 
beginning  of  the  morning  twilight,  by  the  disappearance  of  faint 
stars  situated  in  the  vicinity  of  the  eastern  horizon.  It  has  been 
ascertained  from  numerous  observations,  that,  at  the  beginning  of 
the  morning  and  end  of  the  evening  twilight,  the  sun  is  about 
18°  below  the  horizon. 

353.  The  latitude  of  the  place  and  the  sun^s  declination  being 
given,  to  find  the  time  of  the  hegiymhig  or  end  of  twilight. 

The  zenith  distance  of  the  sun  at  the  beginning  of  morning 
or  end  of  evening  twilight,  is  90°  +  18° :  wherefore,  we  may 
solve  this  problem  by  means  of  equations  (81)  and  (82),  taking  R 
=  18°. 

If  the  time  of  the  commencement  of  morning  twilight  be  sub- 
tracted from  the  time  of  sunrise,  the  remainder  will  be  the  dura- 
tion of  twilight. 

At  the  latitude  49°,  the  sun  at  the  time  of  the  summer  sol- 
stice is  only  18°  below  the  horizon,  at  midnight ;  for,  the  alti- 
tude of  the  pole  at  a  place  the  latitude  of  which  is  49°,  dif- 
fers only  18°  from  the  polar  distance  of  the  sun  at  this 
epoch.  At  this  latitude,  therefore,  twilight  will  continue  all 
night,  at  the  summer  solstice.  This  will  be  true  for  a  still 
stronger  reason  at  greater  latitudes. 


142  ASTRONOIMY. 

354.  Tlie  duration  of  twilight  varies  with  the  latitude  of  the 
place  and  with  the  time  of  the  year.  At  all  places  in  the  north- 
ern hemisphere,  the  summer  are  longer  than  the  winter  twi- 
lights; and  the  longest  twilights  take  place  at  the  summer  sol- 
stice ;  while  the  shortest  occur  when  the  sun  has  a  small 
southern  declination,  different  for  each  latitude.*  The  summer 
twilights  increase  in  length  from  the  equator  northward. 

355.  At  the  Pole  twilight  commences  about  a  month  and  a  half 
before  the  sun  appears  above  the  horizon,  and  lasts  about  a  month 
and  a  half  after  he  has  disappeared. 

The  Seasons. 

356.  The  amount  of  heat  received  from  the  sun  in  the  course 
of  24  hours,  depends  upon  two  particulars,  the  time  of  the  sun's 
continuance  above  the  horizon,  and  the  obliquity  of  his  rays  at 
noon.  By  reason  of  the  obliquity  of  the  ecliptic,  both  of  these  cir- 
cumstances vary  materially  in  the  course  of  the  year ;  whence 
arises  a  variation  of  temperature  or  a  change  of  seasons. 

357.  The  tropics  and  the  arctic  circles  divide  the  earth  into 
five  parts  called  Zones,  throughout  each  of  which  the  yearly 
change  of  the  temperature  is  occasioned  by  a  similar  change  in 
the  circumstances  upon  which  it  depends. 

The  part  contained  between  the  two  tropics,  is  called  the 
Torrid  Zone ;  the  two  parts  between  the  tropics  and  polar 
circles,  are  called  the  Temperate  Zones  ;  and  the  other  two  parts, 
within  the  polar  circles,  are  called  Frigid  Zones. 

358.  At  all  places  in  the  north  temperate  zone  the  sun  will 
always  pass  the  meridian  to  the  south  of  the  zenith  ;  for,  the  lati- 
tudes of  all  such  places  exceed  23^°,  the  greatest  declination  of 
the  sun.  The  meridian  zenith  distance  will  be  greatest  at  the 
winter  solstice,  when  the  sun  has  its  greatest  southern  declina- 


*  The  duration  of  shortest  twilight  is  given  by  the  following  formula 

sin   9° 


cos  lat. 
Twice  the  angle  a,  converted  intotime,  expresses  the  duration  of  shortest  twilight. 
To  findthe  sun's  declination  at  the  time  of  shortest  twilight,  we  have, 

sin  dec.  =  —  tang  9°  sin  lat. 
(For  the  investigation  of  this  and  the  preceding  formula,  see  Gummerc's  Astron- 
omy, pages  87  and  88.) 


THK    SEASONS.  143 

tion,  and  least  at  the  summer  solstice,  when  the  sun  has  its 
greatest  northern  declination  ;  and  it  will  vary  continually  be- 
tween the  values  which  obtain  at  these  epochs.  The  day  will  be 
loujo^est  at  the  summer  solstice,  and  the  shortest  at  the  winter  sol- 
stice, and  will  vary  in  length  progressively  from  the  one  date  to 
the  other. 

We  infer,  therefore,  that  throughout  the  zone  in  question  the 
greatest  amount  of  heat  will  be  received  from  the  sun  at  the  sum- 
mer  solstice,  and  the  least  at  the  winter  solstice  ;  and  that  the 
amount  received  will  gradually  increase,  or  decrease,  from  one 
of  these  epochs  to  the  other.  The  solstices  are  not,  however,  the 
epochs  of  maximum  and  minimum  temperature,  but  are  found 
from  observation  to  precede  these  by  about  a  month.  The  reason 
of  this  circumstance  is,  that  the  earth  continues  for  a  month,  or 
thereabouts,  after  the  summer  solstice,  to  receive  during  the  day 
more  heat  than  it  loses  during  the  night,  and  for  about  the  same 
length  of  time  after  the  winter  solstice,  continues  to  lose  during 
the  night  more  heat  than  it  receives  during  the  day. 

359.  Within  the  torrid  zone  the  length  of  the  day  varies  after 
the  same  manner  as  in  the  temperate  zone,  though  in  a  less  de- 
gree ;  but  the  motion  of  the  sun  with  respect  to  the  zenith,  is 
diiferent.  At  all  places  in  the  torrid  zone  the  sun  passes  the 
meridian  during  a  certain  portion  of  the  year  to  the  south  of  the 
zenith,  and  during  the  remaining  portion  to  the  north  of  it ;  for, 
all  places  so  situated  have  their  zeniths  between  the  tropics  in 
the  heavens,  and  the  sun  moves  from  one  tropic  to  the  other,  and 
back  again  to  its  original  position,  in  a  tropical  year.  Through- 
out the  torrid  zone,  therefore,  the  sun  will  be  in  the  zenith  twice 
in  the  course  of  the  year,  and  will  be  at  its  maximum  distance 
from  it  on  the  one  side  and  the  other,  at  the  solstices. 

An  inhabitant  of  the  equator  or  its  vicinity,  will  have  summer 
at  the  two  periods  when  the  sun  is  in  the  zenith,  and  winter,  (or 
a  period  of  minimum  temperature,)  both  at  the  summer  and  win- 
ter solstice.  Near  the  tropics,  there  will  be  but  little  variation  in 
the  daily  amount  of  heat  received,  during  the  period  that  the  sun 
is  north  of  the  zenith. 

360.  At  the  frigid  zone  a  new  cause  of  a  change  of  temperature 
exists ;  the  sun  remains  continually  above  the  horizon,  for  a 
greater  or  less  number  of  days  about  the  summer  solstice,  and 


144  ASTllONOMV. 

continually  below  it  for  the  same  number  of  days   about  the 
winter  solstice. 

361.  The  amount  of  the  yearly  variation  of  temperature  in- 
creases with  the  latitude  of  the  place  ;  for,  the  greater  is  the  lati- 
tude, the  greater  will  be  the  variation  in  the  length  of  the  day. 
Also,  the  mean  yearly  temperature  is  lower,  as  we  recede  from 
the  equator  and  approach  the  poles ;  for,  since  the  sun  is,  in  the 
course  of  the  year,  the  same  length  of  time  above  the  horizon, 
at  all  places,  the  mean  yearly  temperature  must  depend  altogether 
upon  the  mean  obliquity  of  the  sun's  rays  at  noon,  and  this  in- 
creases with  the  latitude. 

362.  The  yearly  change  in  the  sun's  distance  from  the  earth, 
has  but  little  effect  in  producing  a  variation  of  temperature  upon 
the  earth's  surface.  The  change  of  its  heating  power  from  this 
cause  amounts  to  no  more  than  y^j. 

363.  It  is  important  to  observe,  that,  although  in  the  main  cli- 
mate varies  with  the  latitude  after  the  manner  explained  in  the 
foregoing  articles,  it  is  still  dependent  more  or  less  upon  local 
circumstances,  such  as  the  vicinity  of  lakes,  seas,  and  mountains, 
prevailing  winds  of  some  particular  direction,  <fec. 

364.  In  the  north  temperate  zone,  Spring,  Suinmer,  Autumn^ 
and  Winter,  the  four  seasons  into  which  the  year  is  divided,  are 
considered  as  respectively  commencing  at  the  times  of  the  Yer- 
nal  Equinox,  /Summer  Solstice,  Autum,?ial  Equinox,  and  Win- 
ter Solstice. 

365.  Let  V  (Fig.  50)  represent  the  vernal,  and  A  the  autumnal 
equinox ;  S  the  summer,  and  W  the  winter  solstice.  The  perigee 
of  the  sun's  apparent  orbit  is  at  present  about  10°  10'  to  the  east  of 
the  winter  solstice.  Let  P  denote  its  position.  The  lengths  of 
the  seasons  are,  agreeably  to  Kepler's  law  of  areas,  respectively 
proportional  to  the  areas  V  E  S,  SEA,  A  E  W,  and  W  E  V. 
Thus,  the  winter  is  the  shortest  season,  and  the  summer  the 
longest ;  and  spring  is  longer  than  autumn.  Spring  and  sum- 
mer, taken  together,  are  about  7  days  longer  than  auturrm  and 
winter  united. 

366.  Since  the  perigee  of  the  sun's  orbit  has  a  progressive 
motion,  the  relative  lengths  of  the  seasons  must  be  subject  to  a 
continual  variation. 

367.  At  the  beginning  of  the  year  1800,  the  longitude  of  the 


DIMENSIONS    OF    THE    SUN.  145 

Sim's  perigee  was  279°  30'  8".39.  If  from  this  we  take  180°,  the 
longitude  of  the  autumnal  equinox,  the  remainder,  99°  30'  8".39, 
is  the  distance  of  the  perigee  from  the  autumnal  equinox  at  that 
epoch.  The  motion  of  the  perigee  in  longitude  is  at  the  rate  of 
61".52  per  year.  Dividing  99°  30'  8".39  by  61".52,  the  quotient 
is  5822.  Hence  it  appears  that  about  5800  years  anterior  to  the 
year  1800,  the  perigee  coincided  with  the  autumnal  equinox, 
and  the  apogee  with  the  vernal  equinox.  It  is  perhaps  worthy 
of  remark,  that  this  is  about  the  period  at  which  most  chronolo- 
gists  fix  the  first  residence  of  man  upon  the  earth. 

Appearance,  Dimensions,  and  Physical  Constitution  of  the  Sim, 

368.  The  sun  presents  the  appearance  of  a  luminous  circular 
disc.  But  it  does  not  necessarily  follow  from  this  that  its  sur- 
face is  really  flat ;  for,  such  is  the  appearance  of  all  globular 
bodies,  when  viewed  at  a  great  distance.  It  is  ascertained  from 
observations  with  the  telescope,  that  the  sun  has  a  rotatory  mo- 
tion :  this  being  the  fact,  its  surface  must  in  reality  be  of  a  spheri- 
cal form  ;  for,  otherwise,  it  would  not,  in  presenting  all  its  sides, 
always  appear  under  the  form  of  a  circle. 

369.  The  sun's  real  diameter  is  determined  from  his  apparent 
diameter  and  horizontal  parallax.  Let  A  C  B  (Fig.  51)  repre- 
sent the  sun  or  other  heavenly  body,  and  E  the  place  of  the 
earth  ;  and  let  ^  =  A  E  B  the  sun's  apparent  diameter,  c^  =  2  A 
S  his  real  diameter,  D  =  E  S  his  distance  from  the  earth,  and  R 
=  the  radius  of  the  earth.     We  have  from  the  triangle  A  E  S, 

A  S  =  E  S  tang  lAEB,  or,  2AS  =  2ES  tang  i  A  E  B ; 

and  thus,  </  =  2  D  tang  ^  8  ;       "^^ 

R  '^ 

but,  D= .  .  .  (equa.  9)  ;  A  - 

sinH 

whence,  (^  =  2  R  ^"""^  ^  ^  =^  2  R  ^ J  =  2  R  -J-  .  (83). 

'  sin  H  H  2  H    ^     ^ 

The  mean  apparent  diameter  of  the  sun  is  32'  1".8,  and  his 
mean  horizontal  parallax  8". 58.  Accordingly  we  have,  for  the 
real  diameter  of  the  sun, 

19 


146  ASTRONOMY. 

Thus  the  diameter  of  the  sun  is  about  112  times  the  diameter 
of  the  earth.  The  volume  of  the  sun  then  exceeds  that  of  the 
earth  nearly  in  the  proportion  1123  ^q  i^^  qj.  1407168  to  1. 

370.  From  equation  (S3),  we  may  derive  the  proportion 

d.2R::o:2U. 
Thus,  the  real  diameter  of  a  heavenly  body  is  to  the  diameter 
of  the  earth,  as  the  apparent  diameter  of  the  body  is  to  double 
its  horizontal  parallax. 

371.  When  the  sun  is  viewed  with  a  telescope  of  considerable 
power  and  provided  with  colored  glasses,  black  spots  of  an  irreg- 
ular form,  surrounded  by  a  dark  border,  or  penumbra,  are  often 
seen  on  its  disc.  Their  number,  magnitude,  and  position  on  the 
disc,  are  extremely  variable.  In  some  years  they  are  very  frequent, 
and  appear  in  large  numbers ;  in  others,  none  whatever  are 
seen.  In  some  instances,  as  many  as  fifty,  of  various  forms  and 
sizes,  have  been  counted.  Their  absolute  magnitude  is  often 
very  jjreat.  Spots  are  not  unfrequently  seen  that  subtend  an 
angle  of  1'  or  60".  Now,  the  apparent  diameter  of  the  earth  as 
viewed  at  the  distance  of  the  sun,  is  equal  to  double  the  sun's 
horizontal  parallax,  or  17" :  the  breadth  of  such  spots  must 
therefore  exceed  three  times  the  diameter  of  the  earth,  or  24,000 
miles.  Spots  two  or  three  times  as  large  as  this  have  been 
seen. 

372.  The  form  and  size  of  the  spots  are  subject  to  rapid  and 
almost  incessant  variations.  When  watched  from  day  to  day, 
or  even  from  hour  to  hour,  they  are  seen  to  enlarge  or  contract, 
and  at  the  same  time  to  change  their  forms.  When  a  spot  dis- 
appears, it  always  contracts  into  a  point,  and  vanishes  before 
the  penumbra.  Some  spots  disappear  almost  immediately  after 
they  become  visible  ;  others  remain  for  weeks,  or  even  months. 

373.  Spots  and  streaks  more  luminous  than  the  general  body 
of  the  sun  are  also  frequently  perceived  upon  parts  of  his  disc, 
especially  in  the  region  of  large  spots,  or  of  extensive  groups  of 
spots.  These  are  called  Facidm.  The  penumbra  which  sur- 
rounds each  black  spot  is  also  abruptly  terminated  by  a  border 
of  light  more  brilliant  than  the  rest  of  the  disc. 

374.  When  the  positions  of  the  spots  on  the  disc  are  observed 
from  day  to  day,  it  is  perceived  that  they  all  have  a  common 
motion  in  a  direction  from  east  to  west.     Some  of  the  spots 


sun's  spots,  and  rotation.  147 

close  up  and  vanish  before  they  reach  the  western  limb  ;  others 
disappear  at  the  western  limb,  and  are  never  afterwards  seen  ;  a 
few,  after  becoming  visible  at  the  eastern  limb,  have  been  seen 
to  pass  entirely  across  the  disc,  disappear  from  view  at  the  west- 
ern limb,  and  re-appear  again  at  the  eastern  limb.  The  time 
employed  by  a  spot  m  traversing  the  sun's  disc,  is  about  14  days. 
The  same  time  is  occupied  in  passing  from  the  western  to  the 
eastern  limb,  while  it  is  invisible.  The  motions  of  the  spots  are 
accounted  for,  in  all  their  circumstances,  by  supposing  that  the 
sun  has  a  motion  of  rotation  from  west  to  east,  around  an  axis 
nearly  perpendicular  to  the  plane  of  the  ecliptic  ;  and  that  the 
spots  are  portions  of  the  solid  body  of  the  sun.  The  truth  of 
this  explanation  of  the  apparent  motions  of  the  sun's  spots,  is 
confirmed  by  the  changes  which  are  observed  to  take  place  in 
the  magnitude  and  form  of  the  more  permanent  spots  during 
their  passage  across  the  disc.  When  they  first  come  into  view 
at  the  eastern  limb,  they  appear  as  a  narrow  dark  streak.  As 
they  advance  towards  the  middle  of  the  disc,  they  gradually 
open  out,  and  increase  in  magnitude ;  and  after  they  have 
passed  the  middle  of  the  disc,  contract  by  the  same  degrees  un- 
til they  are  again  seen  as  a  mere  dark  line  upon  the  western 
limb. 

375.  A  spot  returns  to  the  same  position  on  the  disc  in  about 
27^  days.  This  is  not,  however,  the  precise  period  of  the  sun's 
rotation ;  for,  during  this  time  the  sun  has  apparently  moved 
forward  nearly  a  sign  in  the  ecliptic  ;  the  spot  will  therefore 
have  accomplished  so  much  more  than  a  complete  revolution, 
when  it  is  again  seen  by  an  observer  on  the  earth  in  the  same 
position  on  the  disc. 

376.  The  apparent  position  of  a  spot  with  respect  to  the  sun's 
centre  may  be  accurately  determined,  from  day  to  day,  by  ob- 
serving, when  the  sun  is  crossing  the  meridian,  the  right  as- 
censions and  declinations  both  of  the  spot  and  centre.  From 
three  or  more  observations  of  this  kind  the  period  of  the  sun's 
rotation  and  the  position  of  his  equator  may  be  ascertained. 

377.  The  time  of  the  sun's  rotation  on  his  axis  is  about  25^ 
days  ;  the  inclination  of  his  equator  to  the  ecliptic  7°  30' ;  and 
the  heliocentric  longitude  of  the  ascending  node  of  the  equator 
80°  7'. 


148  ASTRONOMY. 

378.  The  only  theories  relative  to  the  physical  constitution 
of  the  sun,  which  deserve  notice,  are  those  of  Laplace  and  Her- 
schel.  Laplace  supposed  that  the  sun  was  an  immense  globe  of 
solid  matter  in  a  state  of  ignition,  and  that  the  spots  upon  his 
disc  were  lars^e  cavities,  where  there  was  a  temporary  intermis- 
sion in  the  evolution  of  luminous  matter.  Sir  W.  Herschel  was 
of  opinion  that  the  sun  was  an  opake  solid  body,  surrounded  by 
two  atmospheres,  of  which  the  first  was  opake  and  non-lumi- 
nous, and  the  second  luminous.  On  this  hypothesis  the  spots 
are  accounted  for  by  supposing  that  openings  occasionally  take 
place  in  the  atmospheres,  through  which  the  dark  body  of  the 
sun  is  seen.  The  penumbra  is  the  portion  of  the  obscure  at- 
mosphere, situated  immediately  around  the  opening  made  in  it. 
This  theory  seems  to  account  for  all  the  circumstances  of  the 
aspect  and  variation  of  the  form  and  magnitude  of  the  spots, 
which  the  other  does  not  do.  Moreover,  M.  Fourier  has  re- 
marked,  that  the  light  given  out  by  incandescent  gases  is  not 
polarized,  while  that  emitted  by  solids  or  liquids  in  combustion, 
is  ;  and  M.  Arago  has  ascertained  that  the  light  of  the  sun,  in 
effect,  does  not  possess  the  properties  of  polarized  light  in  any 
sensible  degree,  which  seems  to  prove  that  it  emanates  from 
a  gas. 

379.  There  has  been  observed,  in  connection  with  the  sun,  at 
certain  periods  of  the  year,  a  faint  light  that  is  visible  before 
sunrise  and  after  sunset,  to  which  has  been  given  the  name  of 
the  Zodiacal  Lights  from  the  circumstance  of  its  being  mostly 
comprehended  within  the  zodiac.  Its  colour  is  white,  and  its 
apparent  figure  that  of  a  spindle,  the  base  of  which  rests  on  the 
sun,  and  the  axis  of  which  lies  in  the  plane  of  the  sun's  equator ; 
such  as  would  be  the  appearance  of  an  ellipsoid  of  revolution, 
havins:  its  centre  coincident  with  that  of  the  sun,  and  its  trans- 
verse  axis  in  the  plane  of  the  solar  equator.  Its  extent  varies 
with  the  season  of  the  year  and  the  state  of  the  atmosphere  ; 
being  sometimes  more  than  100°,  and  at  other  times  not  more 
than  40°  or  50°.  No  satisfactory  explanation  has  yet  been 
given  of  this  singular  phenomenon.  It  was  long  supposed  to  be 
the  atmosphere  of  the  sun,  but  Laplace  has  shown  that  this  hy- 
pothesis is  opposed  to  the  theory  of  gravitation. 

380.  The  zodiacal  light  is  seen  most  distinctly  in  our  north- 


PHASES    OP    THE    MOON.  149 

erii  climates,  early  in  the  spring  just  after  sunset,  and  early  in 
the  fall  just  before  sunrise.  During  the  month  of  March  it  may 
be  seen  directed  towards  the  star  Aldebaran.  Towards  the 
summer  solstice,  it  is  said  to  be  discernible,  in  a  very  pure  state 
of  the  atmosphere,  both  in  the  morning  and  evening.  The  rea- 
son of  these  variations  in  the  distinctness  of  the  zodiacal  light, 
is  found  in  the  change  of  its  inclination  to  the  horizon  at  the 
time  of  sunset  or  sunrise.  As  its  length  lies  in  the  plane  of  the 
sun's  equator,  its  inclination  to  the  horizon  will  be  different, 
like  that  of  this  plane,  according  to  the  different  positions  of  the 
sun  in  the  ecliptic.  Since  the  sun's  equator  makes  but  a  small 
angle  with  the  ecliptic,  at  sunset,  the  zodiacal  light  will  be  most 
inclined  to  the  horizon,  and  therefore  extend  higher  up  in  the 
heavens,  towards  the  vernal  equinox,  when  the  inclination  of  the 
ecliptic  to  the  horizon  at  sunset  is  at  its  maximum ;  and,  at  sun- 
rise, it  will  be  most  inclined  to  the  horizon  towards  the  autum- 
nal equinox,  when  the  inclination  of  the  ecliptic  to  the  horizon 
at  sunrise  is  the  greatest. 


CHAPTER    XV. 

OF    THE    MOON    AND    ITS    PHENOMENA. 

Phases  of  the  Mooii. 

381.  The  most  conspicuous  of  the  phenomena  exhibited  by 
the  moon,  is  the  periodical  change  that  is  observed  to  take 
place  in  the  form  and  size  of  its  disc.  The  different  appear- 
ances which  the  disc  presents  are  called  the  Phases  of  the 
moon. 

The  phenomenon  in  question  is  a  simple  consequence  of  the 
revolution  of  the  moon  around  the  earth.  Let  E  (Fig.  52)  re- 
present the  position  of  the  earth,  ABC,  &c.  the  orbit  of  the  moon^ 
which  we  will  suppose  for  the  present  to  lie  in  the  plane  of  the 


150  ASTRONOMY. 

ecliptic,  and  E  S  the  direction  of  the  sun.     As  the  distance  of 
the  sun  from  the  earth  is  about  400  times  the  distance  of  the 
moon,  lines  drawn  from  the  sun  to  the  different  parts  of  the 
moon's  orbit,  may  be  considered,  witliout  material  error,  as  par- 
rallel  to  each  other.     If  we  regard  the  moon  as  an  opake  non- 
luminous  body,  that  hemisphere  which  is  turned  towards  the 
sun,  will  continually  be  illuminated  by  him,  and  the  other  will 
be  in  the  dark.      Now,  by  virtue  of  the  moon's  motion,  the  en- 
lightened  hemisphere   is  presented  to  the  earth   under   every 
variety  of  aspect  in  the  course  of  a  synodical  revolution  of  the 
moon.     Thus,  when  the  moon  is  in  conjunction,  as  at  A,  this 
hemisphere  is  turned  entirely  away  from  the  earth,  and  she  is  in- 
visible.    Soon  after  conjunction,  a  portion  of  it  on  the  right  begins 
to  be  seen,  and  as  this  is  comprised  betwen  the  right  half  of  the 
circle  which  limits  the  vision,  and  the  right  half  of  the  circle 
which  separates  the  enlightened  and  dark  hemispheres  of  the 
moon,  called  the  Circle  of  Illumination,  it  will  obviously  pre- 
sent the  appearance  of  a  crescent  with  the  horns  turned  from  the 
sun,  as  represented  at  B.     As  the  moon  advances,  more  and  more 
of  the  enlightened  half  becomes  visible,  and  thus  the  crescent 
enlarg-es,  and  the  eastern  limb  becomes  less  concave.     At  the 
point  C,  90°  distant  from  the  sun,  one  half  of  it  is  seen,  and  the 
disc  is  a  semi-circle,  the  eastern  limb  being  a  right  line.     Beyond 
this  point,  more  than  half  becomes  visible  ;  the  nearer  half  of  the 
circle  of  illumination  falls  to  the  left  of  the  moon's  centre,  as 
seen  from  the  earth,  and  thus  becomes  convex  outwards.     This 
phase  of  the  moon  is  represented  at  D.     When  the  moon  appears 
under  this  shape,  it  is  said  to  be  Gibbous.     In  advancing  towards 
opposition,  the  disc  will  enlarge,  and  the  eastern  limb  become 
continually  more  convex  ;  and  finally  at  opposition,  where  the 
whole  illuminated  face  is  seen  from  the  earth,  it  will  become  a 
full  circle.     From  opposition  to  conjunction,  the  nearer  half  of 
the  circle  of  illumination  will  form  the  right  or  western  limb, 
and  this  limb  will  pass  in  the  inverse  order  through  the  same 
variety  of  forms,  as  the  eastern  limb  in  the  interval  between  con- 
junction and  opposition.     The  diflferent  phases  are  delineated  in 
the  figure. 

382.  The  moon's  orbit  is,  in  fact,  somewhat  inclined  to  the 
plane  of  the  ecliptic,  instead  of  lying  in  it,  as  we  have  sup- 


TIME    OF    NEW    OR    FULL    MOON.  151 

posed ;  but,  it  is  plain  that  its  inclination  cannot  change  the 
order,  nor  the  period  of  the  phases,  and  that  it  can  have  no  other 
effect  than  to  alter  somewhat  the  size  of  the  disc,  at  partic- 
ular angular  distances  from  the  sun.  In  consequence  of  the 
smallness  of  the  inclination,  this  alteration  is  too  slight  to  be 
noticed. 

383.  When  the  moon  is  in  conjunction,  it  is  said  to  be  New 
Moon ;  and  when  in  opposition,  Full  Moon.  At  the  time  be- 
tween new  and  full  moon,  when  the  difference  of  the  longitudes 
of  the  moon  and  sun  is  90°,  it  is  said  to  be  the  First  Quarter. 
And  at  the  corresponding  time  between  full  and  new  moon,  it  is 
said  to  be  the  Last  Quarter.  In  both  these  positions  the  moon 
appears  as  a  semi-circle,  and  is  said  to  be  dichototnized.  The 
two  positions  of  conjnnction  and  opposition,  are  called  Syzigies  ; 
and  those  of  the  first  and  last  quarter.  Quadratures.  The  four 
points  midway  between  the  syzigies  and  quadratures,  are  called 
Octants. 

384.  The  interval  from  new  moon  to  new  moon  again,  is 
called  a  Ltmar  Month,  and  sometimes  a  Lunation. 

The  mean  daily  motion  of  the  sun  in  longitude  is  59'  8".33, 
and  that  of  the  w.oon  13°  10'  35".03  ;  wherefore  the  moon  sepa- 
rates from  the  sun  at  the  mean  rate  of  12°  11'  26". 70  per  day  ; 
and  hence,  to  find  the  mean  length  of  a  lunar  month,  we  have 
the  proportion, 

12°  11'  26".70  :  Id.  :  :  360°  :  x  -  29d.  12h.  44m.  2.7s. 

385.  To  determine  the  time  of  m,ean  new  or  full  inoon  in 
any  given  month. 

Let  the  mean  longitude  of  the  sun,  and  also  the  mean  longi- 
tude of  the  moon,  at  the  beginning  of  the  year,  be  found,  and  let 
the  former  be  subtracted  from  the  latter  (adding  360°  if  neces- 
sary) ;  the  remainder,  which  call  R,  will  be  the  mean  distance  of 
the  moon  to  the  east  of  the  sun,  at  the  beginning  of  the  year. 

As  the  moon  separates  from  the  sun  at  the  mean  rate  of  12°  11' 

■p 
26".70perday,   — — -_—-_—_  will  express  the  number  of  days 
^        ^'  12°  11'  26".70  ^  ^ 

and  fractions  of  a  day,  which  at  this  epoch  have  elapsed  since  the 

last  new  moon.     This  interval  is  called  the  Astronomical  Epact. 

If  we  subtract  it  from  29d.  12h.  44m.  2.7s.  we  shall  have  the  time 


152  ASTRONOMY. 

of  mean  new  moon  in  January.  This  being  known,  the  time  of 
mean  new  moon  in  any  other  month  of  the  year  results  very 
readily  from  the  known  length  of  a  lunar  month. 

The  time  of  mean  new  moon  in  any  month  being  known,  the 
time  of  mean  full  moon  in  the  same  month  is  obtained  by  the 
addition  or  subtraction,  as  the  case  may  be,  of  half  a  lunar 
month. 

This  problem  is  in  practice  most  easily  resolved  with  the  aid 
of  tables.     (See  Problem  XXVII). 

386.  The  time  of  true  new  moon  differs  from  the  time  of  mean 
new  moon,  for  the  same  reasons  that  the  true  longitudes  of  the 
sun  and  moon  differ  from  the  mean.  The  same  is  true  of  the 
time  of  true  full  moon.  For  the  mode  of  computing  the  time 
of  true  new  or  full  moon  from  that  of  mean  new  or  full  moon,  (see 
Problem  XXVII.) 

387.  The  earth,  as  viewed  from  the  moon,  goes  through  the 
same  phases  in  the  course  of  a  lunar  month  that  the  moon  does 
to  an  inhabitant  of  the  earth.  But,  at  any  given  time,  the  phase 
of  the  earth  is  just  the  opposite  to  the  phase  of  the  moon.  About 
the  time  of  new  moon,  the  earth,  then  near  its  full,  reflects  so 
much  light  to  the  moon  as  to  render  the  obscure  part  visible. 

Moon's  Rising,  Setting,  and  Passage  over  the  Meridian. 

388.  To  find  the  time  of  the  meridian  passage  of  the  moon  on 
a  given  day. 

Let  S  and  M  denote,  respectively,  the  right  ascension  of  the 
sun,  and  the  right  ascension  of  the  moon,  at  noon  on  the  given 
day,  and  M,  S  the  hourly  variations  of  the  right  ascension  of  the 
sun  and  moon  :  also  let  t  =  the  required  time  of  the  meridian  pas- 
sage.    At  the  time  t  the  right  ascensions  will  be, 

For  the  moon,     .         .         .     M  -f-  ^  7w, 
For  the  sun,        .         .        .      S  +  ^  s  ; 
and,  as  the  moon  is  on  the  meridian,  the  difference  of  these  arcs 
will  be  equal  to  the  hour  angle  t ;  whence, 

t=M  —  ^-\-t{m  —  s)] 
or,  if  all  the  quantities  be  expressed  in  seconds, 

/  =  M  — S  +  j:.^;^^.  .  .(84). 
3600  ^     '' 


TIME    OF    moon's    MERIDIAN    PASSAGE.  153 

Thus,  we  find  for  the  time  of  the  meridian  passage, 
^_    3600  (M-S)  .g 

3600  — (77^—5)  ■  '  ■  ^     '' 
The  quantities  M,  S,  w,  5,  are,  in  practice,  to  be  taken  from 
ephemerides  of  the  sun  and  moon. 

Example.  What  was  the  time  of  the  passage  of  the  moon's 
centre  over  the  meridian  of  New  York,  on  the  1st  of  August, 
1837? 

When  it  is  noon  at  New  York,  it  is  4h.  56m.  4s.  at  Greenwich. 
Now,  by  the  Nautical  Almanac, 
Aug.  1st,  at  4h.  D's  R.  Ascen.     .     .     8''-  58'"-  36.7^- 
"        at  5h.     "         "...     9       0      38.3 


Ih.  r  56m.  4s.  :  :  2m.    1.6s.  :  Im.  53.6s. 

Aug.  1st,  at  4h.  D's  R.  Ascen.     .     .     8^-  58™-  36.7^^- 
Variation  of  R.  Ascen.  in  56m.  4s.  .  1      53.6 


D 's  R.  Ascen.  at  M.  Noon  at  N.York,   9       0      30.3 

Aug.  1st,  O's  hourly  Variation  of  R.  Ascen.  .     .     .  9.704s. 
Ih.  :  4h.  56m.  4s.  :  :  9.704s.  :  47.8s. 

Aug.  1st,  M.  Noon  atGreenw.,  O's  R.  Asc.  8^-  45™-  31. 5«- 
Variation  of  R.  Ascen.  in  4h.  56m.  4s.  47.8 


O's  R.  Ascen.  at  M.  Noon  at  N.  York,        8     46      19.3   (S) 
D's     "  "  "  9       0      30.3   (M) 


M  —  S  =  14      11.0  =  851.0s. 

Aug.  1st,  M.  Noon  at  Greenw.,  D  's  R.  Asc.  8  'i-  50 '"•  27.7^- 
Auff.2d,       "  «  «  9     38      18.7 


24  )  47      51.0 


Aug.  1st,  D  's  mean  hourly  Varia.  of  R.  Asc.       1      59.6   (w?) 
«        o's       »  "  "  9.7   \s) 


m  —  s  =  l     49.9  =  109.9s. 
20 


154  ASTRONOMY. 

3600 lojr.  3.55630 

M  —  S  =  851.0 log.  2.92993 

3600  —  (m  —  s)  =  3490.1  .     .      ar.  co.  log.  6.45716 

Appar.  time  of  merid.  passage,  14m.  37.8s.  =  877.8s.    log.  2.94339 
Equa.  of  time  by  Almanac,        5      58 

Mean  time  of  merid.  pass.,  Oh.  20m.  36s. 

The  Nautical  Almanac  gives  the  time  of  the  moon's  passage 
over  the  meridian  of  Greenwich  for  every  day  of  the  year.  From 
this,  the  time  of  the  passage  across  the  meridian  of  any  other 
place  may  easily  be  determined,  as  follows  :  subtract  the  time  of 
the  meridian  passage  at  Greenwich  on  the  given  day,  from  that 
on  the  following  day,  and  say,  as  24h. :  the  difference  :  :  the 
longitude  of  the  place  :  a  fourth  term.  This  fourth  term  added 
to  the  time  of  the  meridian  passage  at  Greenwich  on  the  given 
day,  will  give  the  approximate  time  of  the  meridian  passage  on 
the  same  day  at  the  given  place.  The  fourth  term  of  the  pre- 
ceding proportion  may  be  corrected,  and  a  more  exact  result 
obtained  by  stating  the  proportion,  as  24h.  :  24h.  -f-  differ- 
ence above  mentioned  :  :  fourth  term  in  question  :  the  same 
corrected. 

389.  Since  the  moon  has  a  motion  with  respect  to  the  sun,  the 
time  of  its  rising  and  setting  must  vary  from  day  to  day.  When 
first  seen  after  conjunction,  it  will  set  soon  after  the  sun.  After 
this  it  will  set  (at  a  mean)  about  one  hour  later  every  succeeding 
night.  At  the  first  quarter,  it  will  set  about  midnight ;  and  at  full 
moon,  will  set  about  sunrise  and  rise  about  sunset.  During  this 
interval  it  will  rise  in  the  day  time,  and  all  along  from  sunrise  to 
sunset.  From  full  to  new  moon,  it  will  rise  at  night,  and  set 
during  the  day ;  and  the  time  of  the  rising  and  setting  will  be 
about  an  hour  later  on  every  succeeding  night  and  day ;  thus,  at 
the  last  quarter  it  will  rise  about  midnight  and  set  about 
midday. 

390.  To  find  the  time  of  the  moon^s  rising  or  setting  on  any 
given  day. 

Compute  the  moon's  semi-diurnal  arc  from  equation  (82),  or 
(80),  according  as  it  is  the  time  of  the  apparent  rising  or  setting, 
or  the  time  of  the  true  rising  or  setting,  that  is  desired.     Correct 


ROTATION    AND    LIBRATIONS    OF    THK    MOON.  155 

it  for  the  moon's  change  of  right  ascension  in  the  interval  between 
the  moon's  passage  over  the  meridian  and  setting,  by  the  follow- 
ing proportion,  24h.  :  24  +  m  —  S  (Art.  388)  :  :  semi-diurnal 
arc  :  corrected  semi-diurnal  arc ;  and  add  it  to  the  time  of  the 
moon's  meridian  passage,  found  as  explained  in  Art.  388.  The 
result  will  be  the  time  of  the  moon's  setting  ;  and  if  tins  be  sub- 
tracted from  24  hours,  the  remainder  will  be  the  time  of  the 
moon's  rising. 

In  consequence  of  the  change  of  the  moon's  declination  in  the 
interval  between  its  rising  and  setting,  it  would  be  more  accurate 
to  compute  the  semi-diurnal  arc  separately  for  the  moon's  rising. 
In  computing  the  semi-diurnal  arc  by  equation  (80),  the  declina- 
tion 6  hours  before  or  after  the  meridian  passage  may  be  used  at 
first ;  and  afterwards,  if  a  more  accurate  result  be  desired,  the  cal- 
culation maybe  repeated  with  the  declination  found  for  the  com- 
puted approximate  time.  In  equation  (82),  R  =  refraction  — 
parallax  =  33'  51"  —  57'  1"  (at  a  mean)  =  —  23'  10". 
Rotation  and  Librations  of  the  Moon. 

391.  The  moon  presents  continually  nearly  the  same  face 
towards  the  earth  ;  for,  the  same  spots  are  always  seen  in  nearly 
the  same  position  upon  the  disc.  It  follows,  therefore,  that  it 
rotates  on  its  axis  in  the  same  direction,  and  with  the  same  an- 
gular velocity,  or  nearly  so,  that  it  revolves  in  its  orbit. 

392.  The  spots  on  the  moon's  disc,  although  they  constantly 
preserve  very  nearly  the  same  situations,  are  not,  however,  strictly 
stationary.  When  carefully  observed,  they  are  seen  alternately 
to  approach  and  recede  from  the  edge.  Those  that  are  very  near 
the  edge  successively  disappear  and  again  become  visible.  This 
vibratory  motion  of  the  moon's  spots  is  called  Libration. 

393.  There  are  three  librations  of  the  mcon,  that  is,  a  vibratory 
motion  of  its  spots  from  three  distinct  causes. 

1.  The  moon's  motion  of  rotation  being  uniform,  small  por- 
tions on  its  east  and  west  sides  alternately  come  into  sight  and  dis- 
appear, in  consequence  of  its  unequal  motion  in  its  orbit.  The 
periodical  oscillation  of  the  spots  in  an  easterly  and  westerly  direc- 
tion from  this  cause,  is  called  the  Libration  in  Longitude. 

2.  The  lunar  spots  have  also  a  small  alternate  motion  from 
north  to  south.  This  is  called  the  Libration  in  Latitude.,  and  is 
accounted  for,  by  supposing  that  the  moon's  axis  is  not  exactly 


156  ASTRONOMY. 

perpendicular  to  the  plane  of  its  orbit,  and  that  it  remains  contin- 
ually parallel  to  itself.  On  this  supposition  we  ought  some- 
times to  see  beyond  the  north  pole  of  the  moon,  and  sometimes 
beyond  the  south  pole. 

3.  Parallax  is  the  cause  of  a  third  libration  of  the  moon.  The 
spectator  upon  the  earth's  surface  bein^  removed  from  its  centre, 
the  point  towards  which  the  moon  continually  presents  the  same 
hemisphere,  he  will  see  portions  of  the  moon  a  little  different 
accordino"  to  its  different  positions  above  the  horizon.  The  diur- 
nal motion  of  the  spots  resulting  from  the  parallax,  is  called  the 
Diurnal  or  Parallactic  Libration. 

394.  The  exact  position  of  the  moon's  equator,  like  that  of  the 
sun's,  is  derived  from  accurate  observations  of  the  situations  of 
the  spots  upon  the  disc.  From  calculations  founded  upon  such 
observations,  it  has  been  ascertained  that  the  plane  of  the  moon's 
equator  is  constantly  inclined  to  the  plane  of  the  ecliptic  under 
an  angle  of  1°  30',  and  intersects  it  in  a  line  which  is  always 
parallel  to  the  line  of  the  nodes.  It  follows  from  the  last  men- 
tioned circumstance,  that  if  a  plane  be  supposed  to  pass 
through  the  centre  of  the  moon,  parallel  to  the  ecliptic,  it  will 
intersect  the  plane  of  the  moon's  equator  and  that  of  its  orbit,  in 
the  same  line  in  which  these  planes  intersect  each  other.  The 
plane  in  question  will  lie  between  the  plane  of  the  equator  and 
that  of  the  orbit.  It  will  make  with  the  first  an  angle  of  1°  30', 
and  with  the  second  an  angle  of  5°  9'. 

Dimensions  and  Physical  Constitution  of  the  Moon. 

395.  The  phases  of  the  moon  prove  it  to  be  an  opake  spherical 
body.     Its  diameter  is  found  by  means  of  equation  (83),  viz  : 

f/  =  2  R  J_  , 
2H  ' 

where  d  denotes  the  diameter  sought,  R  the  radius  of  the  earth,  5 

the  apparent  diameter  of  the  moon  at  a  given  distance,  and  H  its 

horizontal  parallax  at  the  same  distance. 

The  greatest  equatorial  horizontal  parallax  of  the  moon  is  61' 

24".  and  the  corresponding  apparent  diameter  33'  31"  :  thus  we 

have, 

61 '  24"  ^ 

c^  =  2  R  ^-L^jL  =  2  R  _  (very  nearly)  =  2161  miles. 
33'  31"  11  ^      -^  '^ 


moon's  dimensions  and  physical  constitution.      157 

396.  The  diameter  of  the  moon  being  to  the  diameter  of  the 
earth  as  3  to  11,  the  surface  of  the  moon  is  to  the  surface  of  the 
earth  as  3^  to  11'^,  or  as  1  to  13  ;  and  the  volume  of  the  moon  is  to 
the  volume  of  the  earth  as  3^  to  IP,  or  as  1  to  49. 

397.  When  the  moon  is  viewed  with  a  telescope,  the  edge  of 
the  disc,  which  borders  upon  the  dark  portion  of  the  face,  is  seen  to 
be  very  irregular  and  serrated.  It  is  hence  inferred  that  the 
surface  of  the  moon  is  diversified  with  mountains  and  valleys. 
The  truth  of  this  inference  is  confirmed,  by  the  fact,  that  bright 
insulated  spots  are  frequently  seen  on  the  dark  part  of  the  face 
near  the  edge  of  the  disc,  which  gradually  enlarge  until  they  be- 
come united  to  it.  These  briglit  spots  are  doubtless  the  tops  of 
mountains  illuminated  by  the  sun,  while  the  surrounding  regions 
that  are  less  elevated,  are  involved  in  darkness.  The  disc  is  also 
diversified  with  spots  of  different  shapes  and  different  degrees  of 
brightness.  The  brighter  parts  are  supposed  to  be  elevated  land 
and  the  dark  to  be  valleys  or  cavities. 

398.  The  number  of  the  lunar  mountains  is  very  great. 
Their  form  and  grouping  is  for  the  most  part,  similar  to  what 
obtains  in  volcanic  districts  of  the  earth :  from  which  it  is  in- 
ferred that  they  are  of  volcanic  origin. 

399.  From  measurements  made  with  the  micrometer,  of  the 
lengths  of  their  shadows,  or  of  the  distance  of  their  summits, 
when  first  illuminated,  from  the  adjacent  boundary  of  the  disc, 
the  heights  of  a  number  of  the  lunar  mountains  have  been  com- 
puted. According  to  Herschel,  the  altitude  of  the  highest  is 
about  If  English  miles.* 

400.  There  seems  to  be  no  large  bodies  of  water  upon  the  sur- 
face of  the  moon,  or,  at  least,  upon  the  hemisphere  which  is 
turned  towards  the  earth  ;  for,  the  boundary  of  the  illuminated 
hemisphere  is  in  all  its  positions  irregular  throughout,  whereas, 
if  it  ever  fell  upon  any  large  bcdy  of  water,  it  would,  for  the  ex- 
tent of  it,  be  an  unbroken  and  regular  curve. 

The  moon  also  has  no  atmosphere,  or,  if  it  has  any,  it  is  so  rare 
as  not  sensibly  to  diminish  or  refract  the  light  of  the  stars,  passing 


*  Schroeter  makes  tlie  elevation   of  some  of  the  luuar  mountains  to  exceed  5 
miles. 


158  ASTRONOMY. 

through  it ;  for,  when  a  star  experiences  an  occultation  from 
the  moon,  it  does  not  disappear  until  the  body  of  the  moon 
reaches  it,  and  the  duration  of  the  occultation  is  as  it  is  com- 
puted, without  making  any  allowance  for  the  refraction  of  the 
atmosphere. 


CHAPTER     XVI. 

ECLIPSES    OF    THE    SUN    AND    MOON. OCCULTATIONS    OF 

THE    FIXED    STARS. 

401.  An  eclipse  of  a  heavenly  body  is  a  privation  of  its  light, 
occasioned  by  the  interposition  of  some  opake  body  between  it 
and  the  eye,  or  between  it  and  the  sun.  Eclipses  are  divided, 
with  respect  to  the  objects  eclipsed,  into  eclipses  of  the  sun,  of 
the  moon,  and  of  the  satellites  (Art.  304)  ;  and,  with  respect  to 
circumstances,  into  total,  partial,  annular,  and  central.  A 
total  eclipse  is  one  in  which  the  whole  disc  of  the  luminary  is 
darkened  ;  a  partial  one  is  when  only  a  part  of  the  disc  is  dark- 
ened. In  an  annular  eclipse  the  whole  is  darkened,  except  a 
ring  or  annulus,  which  appears  round  the  dark  part  like  an  illu- 
minated border  ;  the  definition  of  a  central  eclipse  will  be  given 
in  another  place. 

Eclipses  of  the  Moon. 

402.  An  eclipse  of  the  moon  is  occasioned  by  an  interposition 
of  the  body  of  the  earth  directly  between  the  sun  and  moon,  and 
thus  intercepting  the  light  of  the  .^'in  ;  or  the  moon  is  eclipsed 
when  it  passes  through  part  of  the  shadow  of  the  earth,  as  pro- 
jected from  the  sun.  Hence  it  is  obvious  that  lunar  eclipses 
can  happen  only  at  the  time  of  full  moon,  for  it  is  then  only 
that  the  earth  can  be  between  the  moon  and  the  sun. 

403.  Since  the  sun  is  much  larger  than  the  earth,  the  shadow 
of  the  earth  must  have  the  form  of  a  cone,  the  length  of  which 


ECLIPSES    OF    'IHE    MOON. — EARTH's    SHADOW.  159 

will  depend  on  the  relative  magnitudes  of  the  two  bodies  and 
their  distance  from  each  other.  Let  the  circles  A  G  B,  ag  b, 
(Fig.  53),  be  sections  of  the  sun  and  earth  by  a  plane  passing 
through  their  centres  S  and  E  ;  A  «,  B  6  tangents  to  these  cir- 
cles on  the  same  side,  and  A  c?,  B  c  tangents  on  different  sides. 
The  triangular  space  a  C  b  will  be  a  section  of  the  earth's  sha- 
dow or  Umbra,  as  it  is  sometimes  called.  The  line  E  C  is 
called  the  Axis  of  the  Shadow.  If  we  suppose  the  line  c  p  io 
revolve  about  E  C,  and  form  the  surface  of  the  frustum  of  a 
cone,  of  which  j)  c  d  g  is  a  section,  the  space  included  within 
that  surface  and  exterior  to  the  umbra,  is  called  the  Penumbra. 
It  is  plain,  that  points  situated  within  the  umbra  will  receive  no 
light  from  the  sun  ;  and  that  points  situated  within  the  penum- 
bra will  receive  light  from  a  portion  of  the  sun's  disc,  and  from 
a  greater  portion  the  more  distant  they  are  from  the  umbra. 

404.  To  find  the  length  of  the  eartKs  shadow. 

Let  L  =  the  length  of  the  shadow ;  R  =  the  radius  of  the  earth  ; 
5  =  sun's  apparent  semi-diameter ;  and  p  =  sun's  parallax.  The 
right  angled  triangle  E  a  C  (Fig.  54)  gives, 

EC= 5.^^ 

tang  E  C  a 

Ea=:R;  andECa=SEA  —  EAC  =  6—p  ;  whence, 

L  = \ .  .  .  (86.) 

tftftg  [6  — p)  ' 

As  the  angle  {h  —  p)  is  only  about  16',  it  will  differ  but  little 
from  its  tangent,  and  therefore, 

L  =  R (nearly)  ; 

0- — p 

or,  if  h  and  p  be  expressed  in  seconds, 
206264".8 
V 

The  shadow  will  obviously  be  the  shortest  when  the  sun  is  the 
nearest  to  the  earth.  We  then  have  5  =  16'  18",  and  p  =  9", 
which  gives  L  =  213  R.  The  greatest  distance  of  the  moon  is  a 
little  less  than  64  R.  It  appears  then,  that  the  eartKs  shadow 
always  extends  to  more  than  three  times  the  distance  of  the 
moon. 

405.  Let  k  BJ  ^  be  a  circular  arc,  described  about  E  the  cen- 


L  =  R  — — - —  (nearly)  .  .  .  (87). 


160  ASTRONOMY. 

tre  of  the  earth,  and  with  a  radius  equal  to  the  distance  between 
the  centres  of  the  earth  and  moon  at  the  time  of  opposition. 
The  angle  M  E  t?*,  the  apparent  semi-diameter  of  a  section  of 
the  earth's  shadow,  made  at  the  distance  of  the  moon's  centre,  is 
called  the  ^emi-diameter  of  the  EartWs  Shadoiv.  And  the 
angle  M  E  /?,  the  apparent  semi-diameter  of  a  section  of  the 
pennmbra,  at  the  same  distance,  is  called  the  Semi-diameter  of 
the  Pemimbra. 

406.  Were  the  plane  of  the  moon's  orbit  coincident  with  the 
plane  of  the  ecliptic,  there  would  be  a  lunar  eclipse  at  every  full 
moon  ;  but,  as  it  is  inclined  to  it,  an  eclipse  can  happen  only 
when  the  full  moon  takes  place  either  in  one  of  the  nodes  of  the 
moon's  orbit,  or  so  near  it  that  the  moon's  latitude  does  not  ex- 
ceed the  sum  of  the  apparent  semi-diameters  of  the  moon  and  of 
the  earth's  shadow. 

To  determine  the  distance  from  the  node,  beyond  which  there 
can  be  no  eclipse,  we  must  ascertain  the  semi-diameter  of  the 
earth's  shadow.  Let  this  be  denoted  by  A,  and  let  P=  the  moon's 
parallax, 

M  E  m  =  E  m  a  —  E  C  w  (Fig.  54) ; 
but  E  m  «  =  P  ;  and  E  C  m  =  5  —  p  (Art.  404) ;  therefore, 
MEw  =  A  =  P-fp  — 5  .  .  .  (88). 

407.  The  semi-diameter  of  the  shadow  is  the  least  when  the 
moon  is  in  its  apogee  and  the  sun  is  in  its  perigee,  or  when  P 
has  its  maximum,  and  6  its  minimum  value.  In  these  positions 
of  the  moon  and  sun,  P  =  53'  48",  8  =  16'  18",  and  p  =  9".  Sub- 
stituting, we  obtain  for  the  least  semi-diameter  of  the  earth's 
shadow  37'  39",  and  for  its  least  diameter  1°  15'  18".  The 
greatest  apparent  diameter  of  the  moon  is  33'  31".  Whence  it 
appears,  that  the  diameter  of  the  earth^s  shadow  is  always  more 
than  twice  the  diameter  of  the  moon. 

The  mean  values  of  P,  />,  and  6  are  respectively  57'  1",  and 
16'  1" ;  which  gives  for  the  mean  semi-diameter  of  the  earth's 
shadow  41'  9". 

408.  If  to  P  -f  7>  —  ^,  the  semi-diameter  of  the  earth's  shadow, 
we  add  d^  the  semi-diameter  of  the  moon,  the  sum  V  -{-j)  -\-  d  —  5 
will  express  the  greatest  latitude  of  the  moon  in  opposition,  at 
which  an  eclipse  can  happen. 


earth's  shadow.  161 

It  is  easy  for  a  given  value  of  P  +/>  +  c^  —  ^j  and  for  a  given 
inclination  of  the  moon's  orbit,  to  determine  within  what  dis- 
tance from  the  node  the  moon  must  be,  in  order  that  an  eclipse 
may  take  place.  By  taking  the  least  and  greatest  inclinations 
of  the  orbit,  the  greatest  and  least  values  of  P  +7^  +  ^  —  5,  and 
also  taking  into  view  the  inequalities  in  the  motions  of  the  sun 
and  moon,  it  has  been  found,  that  when  at  the  time  of  mean 
full  moon,  the  difference  of  the  mean  longitudes  of  the  moon  and 
node  exceeds  13°  21',  there  cannot  be  an  eclipse  ;  but  when  this 
difference  is  less  than  7°  4.7'  there  must  be  one.  Between  7°  47' 
and  13°  21'  the  happening  of  the  eclipse  is  doubtful.  These 
numbers  are  called  the  Lunar  Ecliptic  Limits. 

409.  To  determine  at  what  full  moons  in  the  course  of  any 
one  year  there  will  be  an  eclipse,  find  the  time  of  each  mean  full 
moon  (Art.  385) ;  and  for  each  of  the  times  obtained  find  the 
mean  longfitude  of  the  sun,  and  also  of  the  moon's  node,  and 
compare  the  difference  of  these  with  the  lunar  ecliptic  limits. 
Should,  however,  the  difference,  in  any  instance,  fall  between 
the  two  limits,  farther  calculation  will  be  necessary. 

This  problem  may  be  solved  more  expeditiously  by  means  of 
tables  of  the  sun's  mean  motion  with  respect  to  the  moon's  node. 
(See  Prob.  XXVIII.) 

410.  The  magnitude  and  duration  of  an  eclipse,  depend  upon 
the  proximity  of  the  moon  to  the  node  at  the  time  of  opposition. 
In  order  that  the  centre  of  the  moon  may  be  on  the  same  right 
line  with  the  centres  of  the  sun  and  earth,  or,  in  technical  lan- 
guage, that  a  central  eclipse  may  happen,  the  opposition  must 
take  place  precisely  in  the  node.  A  strictly  central  eclipse,  there- 
fore, seldom,  if  ever,  occurs.  As  the  mean  semi-diameter  of  the 
earth's  shadow  is  41'  9"  (Art.  407),  and  the  mean  hourly  motion 
of  the  moon  with  respect  to  the  sun,  30'  29",  the  mean  duration 
of  a  central  eclipse  would  be  about  2^  h. 

411.  Since  the  moon  moves  from  west  to  east,  an  eclipse  of 
the  moon  must  commence  on  the  eastern  limb,  and  end  on 
the  western. 

412.  In  the  investigations  in  Arts.  404  and  406,  we  have  sup- 
posed the  cone  of  the  earth's  shadow  to  be  formed  by  lines  drawn 
from  the  edge  of  the  sun,  and  touching  the  earth's  surface.  This, 
probably,  is  not  the  exact  case  of  nature  ;  for,  the  duration  of  the 

21 


162  ASTRONOMY. 

eclipse,  and,  by  consequence,  the  apparent  diameter  of  the  earth's 
shadow,  is  found,  by  observation,  to  be  somewhat  greater  than 
would  result  from  this  supposition.  This  circumstance  is  ac- 
counted for,  by  supposing  those  solar  rays,  that,  from  their  direc- 
tion, would  glance  by  and  rase  the  earth's  surface,  to  be  stopped 
and  absorbed  by  the  lower  strata  of  the  atmosphere.  In  such  a 
case  the  conical  boundary  of  the  earth's  shadow  would  be 
formed  by  certain  rays  exterior  to  the  former,  and  would  be 
larger. 

The  moon  in  approaching  and  receding  from  the  earth's  total 
shadow,  or  umbra,  passes  through  the  penumbra,  and  thus  its 
light,  instead  of  being  extinguished  and  recovered  suddenly,  expe- 
riences at  the  beginning  of  the  eclipse  a  gradual  diminution,  and 
at  the  end  a  gradual  increase.  On  this  account  the  times  of  the 
beginning  and  end  of  the  eclipse  cannot  be  noted  with  precision, 
and  in  consequence  astronomers  differ  as  to  the  amount  of  the 
increase  in  the  size  of  the  earth's  shadow  from  the  cause  above 
mentioned.  It  is  the  practice,  however,  in  computing  an  eclipse 
of  the  moon,  to  increase  the  semi-diameter  of  the  shadow  by  a  ^\ 
part ;  or,  which  amounts  to  the  same,  to  add  as  many  seconds  as 
the  semi-diameter  contains  minutes. 

413.  It  is  remarked  in  total  eclipses  of  the  sun,  that  the  moon 
is  not  wholly  invisible,  but  appears  with  a  dull  reddish  light. 

This  phenomenon  is  doubtless  another  effect  of  the  earth's 
atmosphere,  though  of  a  totally  different  nature  from  the  preced- 
ing. Certain  of  the  sun's  rays,  instead  of  being  stopped  and 
absorbed,  are  bent  from  their  rectilinear  course  by  the  refracting 
power  of  the  atmosphere,  so  as  to  form  a  cone  of  faint  light, 
interior  to  that  cone  which  has  been  mathematically  described 
as  the  earth's  shadow,  which  falling  upon  the  moon  renders  it 
visible. 

414.  As  an  eclipse  of  the  moon  is  occasioned  by  a  real  loss  of 
its  light,  it  must  begin  and  end  at  the  same  instant,  and  present 
precisely  the  same  appearance,  to  every  spectator  who  sees  the 
moon  above  his  horizon  during  the  eclipse.  It  will  be  shown 
that  the  case  is  different  with  eclipses  of  the  sun. 

Calculation  of  an  Eclipse  of  the  Moon. 

415.  The  apparent  distance  of  the  centre  of  the  moon  from 
the  axis  of  the  earth's  shadow,  and  the  arcs  passed  over  by  the 


moon's  relative  orbit.  163 

centre  of  the  moon  and  the  axis  of  the  shadow  during  an  echpse 
of  the  moon,  being  necessarily  small,  they  may  without  material 
error  be  considered  as  right  lines.  We  may  also  consider  the 
apparent  motion  of  the  sun  in  longitude,  and  the  motions  of  the 
moon  in  longitude  and  latitude,  as  uniform  during  the  eclipse. 
These  suppositions  being  made,  the  calculation  of  the  circum- 
stances of  an  eclipse  of  the  moon  is  very  simple. 

Let  N  F  (Fig.  55)  be  a  part  of  the  ecliptic,  N  the  moon's  as- 
cending node,  N  L  a  part  of  the  moon's  orbit,  C  the  centre  of  a 
section  of  the  earth's  shadow  at  the  moon,  C  K  perpendicular  to 
N  F  a  circle  of  latitude,  and  C  the  centre  of  the  moon  at  the 
instant  of  opposition  :  then  C  C,  which  is  the  latitude  of  the 
moon,  in  opposition,  is  the  distance  of  the  centres  of  the  shadow 
and  moon  at  that  time.  The  moon  and  shadow  both  have  a 
motion,  and  in  the  same  direction,  as  from  N  towards  F  and  L. 
It  is  the  practice,  however,  to  regard  the  shadow  as  stationary,  and 
to  attribute  to  the  moon  a  motion  equal  to  the  relative  motion  of 
the  moon  and  shadov/.  The  orbit  that  would  be  described  by 
the  moon's  centre  if  it  had  such  a  motion,  is  called  the  Relative 
Orbit  of  the  moon.  Inasmuch  as  the  circumstances  of  the  eclipse 
depend  altogether  upon  the  relative  motion  of  the  moon  and 
shadow,  this  mode  of  proceeding  is  obviously  allowable. 

As  the  shadow  has  no  motion  in  latitude,  the  relative  motion 
of  the  moon  and  shadow  in  latitude  will  be  equal  to  the  moon's 
actual  motion  in  latitude.  And  since  the  centre  of  the  earth's 
shadow  moves  in  the  plane  of  the  ecliptic  at  the  same  rate  as  the 
sun,  the  relative  motion  of  the  moon  and  shadow  in  lono;itude 
will  be  equal  to  the  difference  between  the  motions  of  the  sun 
and  moon  in  longitude.  We  obtain,  therefore,  the  relative  posi- 
tion of  the  centres  of  the  moon  and  shadow  at  any  interval  t, 
following  opposition,  by  laying  off  C  m  equal  to  the  difference 
of  the  motions  of  the  sun  and  moon  in  longitude  in  this  interval, 
through  in  drawing  m  M  perpendicular  to  N  F,  and  cutting  off 
m  M  equal  to  the  latitude  at  opposition  plus  the  motion  in  lati- 
tude in  the  interval  t :  M  will  be  the  position  of  the  moon's  cen- 
tre in  the  relative  orbit,  the  centre  of  the  shadow  being  supposed 
to  be  stationary  at  C.  As  the  motion  of  the  sun  in  longitude, 
and  of  the  moon  in  longitude  and  latitude,  is  considered  uniform, 
the  ratio  of  C  m'  (=  C  w,  the  difference  between  the  motions  of 


164  ASTRONOMY. 

the  sun  and  moon  in  longitude,)  to  M  m'  the  moon's  motion  in 
latitude,  is  the  same,  whatever  may  be  the  length  of  the  interval 
considered.  It  follows,  therefore,  that  the  relative  orbit  of  the 
moon  N'  C  M  is  a  right  line. 

416.  The  relative  orbit  passes  through  C,  the  place  of  the 
moon's  centre  at  opposition :  its  position  will  therefore  be 
known,  if  its  inclination  to  the  ecliptic  be  found.  Now,  we 
have, 

, .  M  m'  moon's  motion  in  latitude 

tan  inciina.  = 


C  m'     moon's  mot.  in  long.  —  sun's  mot.  in  long. 

417.  The  following  data  are  requisite  in  the  calculation  of 
the  circumstances  of  a  lunar  eclipse  : 

T  =  time  of  opposition. 
M  —  moon's  hourly  motion  in  longitude. 
n  —  moon's  hourly  motion  in  latitude. 
7n  =  sun's  hourly  motion  in  longitude. 
X  =  moon's  latitude  at  opposition. 
d  —  moon's  semi-diameter. 
6  =  sun's  semi-diameter. 
P  =  moon's  horizontal  parallax. 
J)  =  sun's  horizontal  parallax. 
s  =  semi-diameter  of  earth's  shadow. 
I  =  inclination  of  relative  orbit. 
h  —  moon's  hourly  motion  on  relative  orbit. 

T,  M,  7i,  m,  X,  c/,  (5,  P,  and  /j,  are  derived  from  Tables  of  the 
sun  and  moon.     (See  Problems  IX  and  XIV.) 

The  quantities  5,  I  and  A,  may  be  determined  from  these, 

5=P+p  — 6-f  gV  (P+i^  — ^)  (Arts. 406 and  412)  ..  .  (89); 
tang  I  =  - (Art.  416)  .  .  .  (90). 

The  triangle  C  M  m'  gives, 

cos  M  C  m'  cos  I  ^     ' 

418.  The  above  quantities  being  supposed  to  be  known,  let  N' 
C  F  (Fig.  56)  represent  the  ecliptic,  and  C  the  stationary  centre 
of  the  earth's  shadow.  Let  C  C  =  X,  and  let  N'  C  L'  represent 
the  relative  orbit  of  the  moon.     We  here  suppose  the  moon  to  be 


TIMES    OF    THE    PHASES    OF    THE    ECLIPSE.  165 

north  of  the  echptic  at  the  time  of  opposition,  and  near  its  ascend- 
ing node  ;  when  it  is  south  of  the  ecHptic,  X  is  to  be  laid  off  below 
N'  C  F,  and  when  it  is  approaching-  either  node,  the  relative  orbit 
is  inclined  to  the  right.  Let  the  circle  K  F  K'  R,  described  about 
the  centre  C,  represent  the  section  of  the  earth's  shadow  at  the 
moon;  and  let/,/'  and  g,  g'  be  the  respective  places  of  the 
moon's  centre,  at  the  beginning  and  end  of  the  eclipse,  and  at 
the  beginning  and  end  of  the  total  eclipse.  C/=  C  f  =  s  -{-  d, 
and  Cg=Cg'  =  s~d.  Draw  C  M  perpendicular  to  N'  C  L', 
and  M  will  represent  the  place  of  the  moon's  centre  when  near- 
est the  centre  of  the  shadow  :  it  will  also  be  its  place  at  the  mid- 
dle of  the  eclipse  ;  for,  since  C/=  C/,  and  C  M  is  perpendicu- 
lar to  N' C'/,  C'/=C'/. 

419.  Middle  of  the  eclipse. 

The  time  of  opposition  being  known,  that  of  the  middle  of  the 
eclipse  will  become  known  when  we  have  found  the  interval 
{x)  employed  by  the  moon  in  passing  from  M  to  C.     Now, 

(expressed  in  parts  of  an  hour)  x  —  — -—  ; 

and  in  the  right  angled  triangle  C  C  M,  we  have  C  C  =  X,  and 

^  C  C  M  =  Z.  C  N'  C  =  I ;  and  therefore  M  C  =  X  sin  I ;  whence, 

by  substitution, 

XsinI      X  sin  I   ,           ni\      ^  sin  I  cos  I 
X  =  — - —  =  — (equa.  91)  =  — — • ; 

cos  I 

or,  (expressed  in  seconds)  x  —  — — — : X  sin  I .  .  .  (92). 

Hence,  if  M  =  time  of  middle,  we  have, 

M  =  T±:r  =  TT  ?5?2!l^^  X.  sin  I  .  .  .  (93). 
M  — m  ^     ^ 

It  is  obvious  that  the  upper  sign  is  to  be  used  when  the  lati- 
tude is  increasifig,  and  the  loiver  sign  when  it  is  decreasing. 

The  distance  of  the  centre  of  the  moon  from  the  centre  of  the 
shadow  at  the  middle  of  the  eclipse, 

=  C  M  =  C  C  cos  C  C  M  =  X  cos  I  .  .  .  (94). 

420.  Beginning  and  end  of  the  eclipse. 

Let  any  point  I  of  the  relative  orbit  be  the  place  of  the  moon's 
centre  at  the  time  of  any  given  phase  of  the  eclipse.     Let  t  =  the 


165  ASTRONOMY. 

interval  of  time  between  the  given  phase  and  the  middle  ;  and 

k  =  C  I,  the  distance  of  the  centres  of  the  moon  and  shadow.     In 

the  interval  t  the  moon's  centre  will  pass  over  the  distance  M  I ; 

hence, 

^  _  M  Z  _  M  Z  cos  I . 

7i"~   M  — m  ' 


but,         M  Z  =  v/  cl'  —  CTf  =  ^  ^*'  —  >^'  cos'  I  (equa.  94), 

and  therefore,  ^  ^  w^ v/ A-^  —  X' cos'T; 

JVI  —  111  ' 

/•  J  \^     3600s.cosl    ,-— — — -^      ,^^, 

or,  (m  seconds)  t  =  —  v/  (/c+XcosI)  [k—'K  cos  I) . .  (95). 

Let  T'  denote  the  time  of  the  supposed  phase  of  the  eclipse, 

and  M  the  time  of  the  middle  ;  and  we  shall  have, 

T'  =  M  +  ^,  or  T'  =  M  —  t, 

according  as  the  phase  follows  or  precedes  the  middle. 

Now  at  the  beginning  and  end  of  the  eclipse,  we  have, 

A;=C/or  C/'  =  s  +  cZ; 

substituting  in  equation  (95)  we  obtain. 

^,     3600s.  cos  I   ,- J. — ,._. 

<  =  -^ v^  (6-  +  (/  +  X  COS  I)  (5  +  </  —  X  COS  I  •  .  (96)  ; 

t'  being  found,  the  time  of  the  beginning  (B),  and  the  time  of  the 
end  (E),  result  from  the  equations, 

B  =  M— ^',  E  =  M  +  ^'. 

421.  Beginning  and  end  of  the  total  eclipse. 
At  the  beo^inning  and  end  of  the  total  eclipse,  k  =  C  g  =  C  g' 
z=s  —  d;  whence  by  equation  (95), 

3600s.  cos  I 


t"  =  — — V  {j  —  d  +  -k  cosl)  {s—  d  —  X  cosl)  .  .  (97); 

and,  denoting  the  time  of  the  beginning  by  B'  and  the  time  of 
the  end  by  E',  we  have, 

B'  =  M  —  t",  E'  =  M  -f  t". 

422.   Quantity  of  the  eclipse. 

In  a  pjirtial  eclipse  of  the  moon  the  magnitude  or  quantity  of 
the  eclipse  is  measured  by  the  relative  portion  of  that  diameter 
of  the  moon,  which,  if  produced,  would  pass  through  the  centre 
of  the  earth's  shadow,  that  is  involved  in  the  shadow.     The 


QUANTITY  AND  CONSTRUCTION  OF  A  LUNAR  ECLIPSE.       167 

whole  diameter  is  divided  into  twelve  equal  parts,  called  Digits, 
and  the  quantity  is  expressed  by  the  number  of  digits  and  frac- 
tions of  a  digit  in  the  part  immersed.  When  the  moon  passes 
entirely  within  the  shadow,  as  in  a  total  eclipse,  the  quantity  of 
the  eclipse  is  expressed  by  the  number  of  digits  contained  in  the 
part  of  the  same  diameter  prolonged  outwards,  which  is  com- 
prised between  the  edge  of  the  shadow  and  the  inner  edge  of  the 
moon.  Thus  the  number  of  digits  contained  in  S  N  (Fig.  56), 
expresses  the  quantity  of  the  eclipse  represented  in  the  figure. 
Hence,  if  d  =  the  quantity  of  the  eclipse,  we  shall  have, 

Q  =    ^^^     =  12  N  S  ^  12  (N  M  +  M  S)  ^ 

t' 2  N  V        NV  N V 

12  (N  M  +  C  S  —  C  M)  ^  ]  2  (cZ  +  s  —  XcosI . 
N  V  2d  ' 

or,  a  =  6(^+^-XcosI)_  _         ^gg^_ 

If  X  cos  I  exceeds  {  s  -[-  d)  there  will  be  no  eclipse.  If  it  is 
intermediate  between  [s  +  d)  and  (6-  —d  )  there  will  be  a  partial 
eclipse  ;  and  if  it  is  less  than  [s  —  d)  the  eclipse  will  be  total. 

Construction  of  an  Eclipse  of  the  Moon. 
423.  The  times  of  the  different  phases  of  an  eclipse  of  the 
moon  may  easily  be  determined  by  a  geometrical  construction, 
within  a  minute  or  two  of  the  truth.  Draw  a  right  line  N'  F 
(Fig.  57)  to  represent  the  ecliptic ;  and  assume  upon  it  any 
point  C,  for  the  position  of  the  centre  of  the  earth's  shadow  at 
the  time  of  opposition.  Then,  having  fixed  upon  a  scale  of 
equal  parts,  lay  off  C  R  =  M  —  m  the  difference  of  the  hourly 
motions  of  the  sun  and  moon  in  longitude  ;  and  draw  the  per- 
pendiculars C  C  =  X  the  moon's  latitude  in  opposition,  and  R 
L'  =  X  ±  n  the  moon's  latitude  an  hour  after  opposition.  The 
right  line  C  L'  drawn  through  C  and  L',  will  represent  the 
moon's  relative  orbit.  It  should  be  observed,  that  if  the  lati- 
tudes are  south,  they  must  be  laid  off  below  N'  F,  and  that  N'  C 
L'  will  be  inclined  to  the  right  when  the  latitude  is  decreasing. 
With  a  radius  C  R  =  5  (equation  89)  describe  the  circle  E  K  F 
K',  which  will  represent  the  section  of  the  earth's  shadow.  With 
a  radius  =  s  -\-  d,  and  another  radius  =  s  —  d,  describe  about  the 
centre  C  arcs  intersecting  N'  L'  in/,/',  and  g,  g' ;  /and/  will 


168  AS'iUONOMY. 

be  the  places  of  the  moon's  centre  at  the  beginning:  and  end  of 
the  ecHpse,  and  g  and  g'  the  places  at  the  beginning  and  end 
of  the  total  eclipse.  From  the  point  C  let  fall  upon  N'  C  L' 
the  perpendicular  C  M  ;  and  M  will  be  the  place  of  the  moon's 
centre  at  the  middle  of  the  eclipse.  To  render  the  construction 
explicit,  let  us  suppose  the  time  of  opposition  to  be  7h.  23m.  15s. 
At  this  time  the  moon's  centre  will  be  at  C.  To  find  its  place 
at  7h.,  state  the  proportion,  60m.  :  23m.  15s. :  :  moon's  hourly  mo- 
tion on  the  relative  orbit :  a  fourth  term.  This  fourth  term  will 
be  the  distance  of  the  moon's  centre  from  the  point  C  at  7 
o'clock,  and  if  it  be  taken  in  the  dividers  and  laid  off  on  the 
relative  orbit  from  C  backwards  to  the  point  7,  it  will  give  the 
moon's  place  at  that  hour.  This  being  found,  take  in  the  divi- 
ders the  moon's  hourly  motion  on  the  relative  orbit,  and  lay  it 
off  repeatedly,  both  forwards  and  backwards  from  the  point  7, 
and  the  points  marked  off,  8,  9,  10,  6,  5,  will  be  the  moon's 
places  at  those  hours  respectively.  Now,  the  object  being  to 
find  the  times  at  which  the  moon's  centre  is  at  the  points/,/', 
g.  g',  and  M,  let  the  hour  spaces  thus  found  be  divided  into  quar- 
ters, and  these  subdivided  into  5  minute  or  minute  spaces,  and 
the  times  answerinsr  to  the  points  of  division  that  fall  nearest  to 
these  points,  will  be  within  a  minute  or  so  of  the  times  in  ques- 
tion. For  example,  the  point/'  falls  between  9  and  10,  and  thus 
the  end  of  the  eclipse  will  occur  somewhere  between  9  and  10 
o'clock.  To  find  the  number  of  minutes  after  9  at  which  it 
takes  place,  we  have  only  to  divide  the  space  from  9  to  10  into 
four  equal  parts  or  15  minute  spaces,  subdivide  the  part  which 
contains  /  into  three  equal  parts  or  5  minute  spaces,  and  again 
that  one  of  these  smaller  parts  within  which  /'  lies,  into  five 
equal  parts  or  minute  spaces. 

Eclipses  of  the  >Sim. 

424.  An  eclipse  of  the  sun  is  caused  by  the  interposition  of  the 
moon  between  the  sun  and  earth ;  whereby  the  whole,  or  part  of 
the  sun's  light,  is  prevented  from  falling  upon  certain  parts  of  the 
earth's  surface. 

Let  A  G  B  and  agb  (Fig.  58)  be  sections  of  the  sun  and  earth 
by  a  plane  passing  through  their  centres  S  and  E,  A  a,  B  b  tan- 
gents to  the  circles  AGB  and  a  ^  6  on  the  same  side,  and  A  c?,  B  c 
tangents  to  the  same  on  opposite  sides.     The  figure  A  a  6  B  will 


SOLAR  ECLIPTIC  LIMITS.  169 

be  a  section  through  the  axis,  of  a  frustum  of  a  cone  formed  by- 
rays  tangent  to  the  sun  and  earth  on  the  same  side,  and  the  trian- 
gular space  Fed  will  be  a  section  of  a  cone  formed  by  rays 
tangent  on  opposite  sides.  An  eclipse  of  the  sun  will  take  place 
somewhere  upon  the  earth's  surface,  whenever  the  sun  comes 
within  the  frustum  AabB,  and  a  total  or  an  annular  eclipse 
whenever  the  sun  comes  within  the  cone  Fed. 

425.  Let  m  m'  M  (Fig.  58)  be  a  circular  arc  described  about  the 
centre  E,  and  with  a  radius  equal  to  the  distance  of  the  centres  of 
the  moon  and  earth  at  the  time  of  conjunction.  The  angle 
niFiS  is  the  apparent  semi-diameter  of  a  section  of  the  frustum, 
and  m'  E  S  the  apparent  semi-diameter  of  a  section  of  the  cone  at 
the  distance  of  the  moon.  To  find  expressions  for  these  semi- 
diameters  in  terms  of  determinate  quantities,  let  the  first  be 
denoted  by  A,  and  the  second  by  A' ;  and  let  P  -  the  parallax 
of  the  moon,  p  =  the  parallax  of  the  sun,  and  6  —  the  serai-diame- 
ter of  the  sun.     Then  we  have, 

wES=A=mEA-fAES=Ema  — EAm  +  AES; 
or,  A  =  P— ;p  +  <5  .  .  .  (99). 

and     m'ES^m'EB  —  BES  =  Em'c  —  EBm'  — BES; 
or,  A'  =  P  — p  — ^  .  .  .  (100). 

Taking  the  mean  values  of  P,  /j,  and  5  (Art.  407),  we  find  for  the 
mean  value  of  A  1°  12'  53",  and  for  the  mean  value  of  A'  40'  51". 

426.  As  the  plane  of  the  moon's  orbit  is  not  coincident  with 

the  plane  of  the  ecliptic,  an  eclipse  of  the  sun  can  happen  only 

when  conjunction  or  new  moon  takes  place  in  one  of  the  nodes 

of  the  moon's  orbit,  or  so  near  it  that  the  moon's  latitude  does 

not  exceed  the  sum  of  the  semi-diameters  of  the  moon  and  of  the 

luminous  frustum  (Art.  425)  at  the  moon's  orbit.     Thus,  denoting 

the  moon's  semi-diameter  by  d,  and  the  greatest  latitude  of  the 

moon  in  conjunction,  at  which  an  eclipse  can  take  place,  by  L, 

we  have,  *y     , 

L  =P_p+6-f  t^  .  .  .  (101).     ^^l>^:- 

For  a  total  eclipse,  the  greatest  latitude  will  be  equal  to  the 
sum  of  the  semi-diameters  of  the  moon  and  the  luminous  cone- 
Hence,  denoting  it  by  L',  --..  i 

L'  =  P  —p  —  6-\-d...  (102).  ^-mvl; 
In  order  that  an  annular  eclipse  may  take  place,  the  apparent 
22 


170  ASTRONOMY. 

semi -diameter  of  the  moon  must  be  less  than  that  of  the  sun,  and 
the  moon  must  come  at  conjunction  entirely  within  the  himinous 
frustum.  Whence,  if  L"  =  the  maximum  latitude  at  which  an 
annular  eclipse  is  possible,  we  have, 

L"  =  P  —  p  +  (5  —  (/  .  .  .  (103).  Jy  U  t'^ 

427.  Li  the  same  manner  as  in  the  case  of  an  eclipse  of  the 
moon,  it  has  been  found  that  when  at  the  time  of  mean  new 
moon  the  difference  of  the  mean  longitudes  of  the  sun  or  moon 
and  of  the  node,  exceeds  19°  44',  there  cannot  be  an  eclipse  of  the 
sun  ;  but  when  the  difference  is  less  than  13°  33',  there  must  be 
one.     These  numbers  are  called  the  Solar  Ecliptic  Limits. 

428.  In  order  to  discover  at  what  new  moons  in  the  course 
of  a  year  an  eclipse  of  the  sun  will  happen,  with  its  approximate 
time,  we  have  only  to  find  the  mean  lons^itudes  of  the  sun  and 
node  at  each  mean  new  moon  throughout  the  year  (Art.  385), 
and  take  the  difference  of  the  longitudes,  and  compare  it  with 
the  solar  ecliptic  limits.  (For  a  more  direct  method  of  solving 
this  problem,  see  Prob.  XXVIII). 

429.  Eclipses  both  of  the  sun  and  moon  recur  in  nearly  the 
same  order  and  at  the  same  intervals  at  the  expiration  of  a  period 
of  223  lunations,  or  18  years  of  365  days,  and  15  days,*  which 
for  this  reason  is  called  the  Period  of  the  Eclijises.  For,  the 
time  of  a  revolution  of  the  sun  with  respect  to  the  moon's  node  is 
346.619851d.,  and  the  time  of  a  synodic  revolution  of  the  moon 
is  29.5305887d.  These  numbers  are  very  nearly  in  the  ratio 
of  223  to  19.  Thus,  in  a  period  of  223  lunations  the  sun  will 
have  returned  19  times  to  the  same  position  with  respect  to  the 
moon's  node,  and  at  the  expiration  of  this  period,  will  be  in  the 
same  position  with  respect  to  the  moon  and  node,  as  at  its  com- 
mencement. The  eclipses  which  occur  during  one  such  period 
being  noted,  subsequent  eclipses  are  easily  predicted. 

This  period  was  known  to  the  Chaldeans  and  Egyptians,  by 
whom  it  was  called  Saras. 

430.  As  the  solar  ecliptic  limits  are  more  extended  than  the 
lunar,  eclipses  of  the  sun  must  occur  more  frequently  than 
eclipses  of  the  moon. 

»  More  exactly  18  years  (of  3G5  day.s)  plus  15d.  7h.  42m.  29s, 


NUMB12R    OF    ECLIPSES    IN     V    YEAR.  171 

As  to  the  number  of  eclipses  of  both  luminaries,  there  cannot 
be  fewer  than  two,  nor  more  than  seven  in  one  year.  The 
most  usual  number  is  four,  and  it  is  rare  to  have  more  than  six. 
When  there  are  seven  eclipses  in  a  year,  five  are'of  the  sun 
and  two  of  the  moon  ;  and  when  but  two,  both  are  of  the  sun. 
The  reason  is  obvious.  The  sun  passes  by  both  nodes  of  the 
moon's  orbit  but  once  in  a  year,  unless  he  passes  by  one  of  .them 
in  the  beginning  of  the  year,  in  which  case  he  will  pass  by  the 
same  again  a  little  before  the  end  of  the  year,  as  he  returns  to  the 
same  node  in  a  period  of  346  days.  Now,  if  the  sun  be  at  a  little 
less  distance  than  13°  33'  from  either  node  at  the  time  of  new 
moon,  he  will  be  eclipsed,  and  at  the  subsequent  opposition  the 
moon  will  be  eclipsed  near  the  other  node,  and  come  round  to  the 
next  conjunction  before  the  sun  is  13°  33'  from  the  former  node. 
And  when  three  eclipses  happen  about  either  node,  the  like 
number  commonly  happens  about  the  opposite  one  ;  as  the  sun 
comes  to  it  in  173  days  afterward,  and  six  lunations  contain  only 
four  days  more.  Thus  there  may  be  two  eclipses  of  the  sun  and 
one  of  the  moon  about  each  of  the  nodes  ;  and  the  twelfth  luna- 
tion from  the  eclipse  in  the  beginning  of  the  year,  may  gfive  a 
new  moon  before  the  year  is  ended,  which,  in  consequence  of  the 
retrogradation  of  the  nodes,  may  be  within  the  solar  ecliptic  limit ; 
and  hence  there  may  be  seven  eclipses  in  a  year,  five  of  the  sun 
and  two  of  the  moon.  But  when  the  moon  changes  in  either  of 
the  nodes,  she  cannot  be  near  enough  the  other  node,  at  the  next 
full  moon,  to  be  eclipsed,  as  in  the  interval  the  sun  will  move» 
over  an  arc  of  14°  32',  whereas  the  greatest  lunar  ecliptic  limit  is 
but  13°  21',  and  in  six  lunar  months  afterwards  she  will  change 
near  the  other  node  ;  in  this  case  there  cannot  be  more  than  two 
eclipses  in  a  year,  both  of  which  will  be  of  the  sun.  If  the  moon 
changes  at  the  distance  of  a  few  degrees  from  either  node,  then 
an  eclipse  both  of  the  sun  and  moon  will  probably  occur  in  the 
passage  of  that  node  and  also  of  the  other. 

431.  Althouofh  solar  eclipses  are  more  frequent  than  lunar, 
when  considered  with  respect  to  the  whole  earth,  yet  at  any 
given  place  more  lunar  than  solar  eclipses  are  seen.  The  reason 
of  this  circumstance,  is  that  an  eclipse  of  the  sun  (unlike  an 
eclipse  of  the  moon)  is  visible  only  over  a  part  of  a  hemisphere 


172  ASTRONOMY. 

of  the  earth.  To  show  this,  suppose  two  Unes  to  be  drawn  from 
the  centre  of  the  moon  tangent  to  the  earth  at  opposite  points  : 
they  will  make  an  angle  with  each  other  equal  to  double  the 
moon's  horizontal  parallax,  or  of  1°  54'.  Therefore,  should  an 
observer  situated  at  one  of  the  points  of  tangency,  refer  the  cen- 
tre of  the  moon  to  the  centre  of  the  sun,  an  observer  at  the  other 
wouW  see  the  centres  of  these  bodies  distant  from  each  other  an 
ano-le  of  1°  54',  and  their  nearest  limbs  separated  by  an  arc  of 
more  than  1°. 

432.  Instead  of  regarding  an  eclipse  of  the  sun  as  produced  by 
an  interposition  of  the  moon  between  the  sun  and  earth,  as  we 
have  hitherto  considered  it,  we  may  regard  it  as  occasioned  by  the 
moon's  shadow  falling  upon  the  earth.  Fig.  59  represents  the 
moon's  shadow  as  projected  from  the  sun  and  covering  a  portion 
of  the  earth's  surface.  Wherever  the  umbra  falls,  there  is  a  total 
echpse  ;  and  wherever  the  penumbra  falls,  a  partial  eclipse. 

433.  In  order  to  discover  the  extent  of  the  portion  of  the  earth's 
surface  over  which  the  eclipse  is  visible  at  any  particular  time, 
we  have  only  to  find  the  breadth  of  the  portion  of  the  earth  cov- 
ered by  the  penumbral  shadow  of  the  moon  ;  but  we  will  first 
ascertain  the  length  of  the  moon's  shadow.  As  seen  at  the  vertex 
of  the  moon's  shadow,  the  apparent  diameters  of  the  moon  and 
sun  are  equal.  Now,  as  seen  at  the  centre  of  the  earth,  they  are 
nearly  equal,  sometimes  the  one  being  a  little  greater  and  some- 
times the  other.     It  follows,  therefore,  that  the  length  of  the 

"moorCs  shadow  is  about  equal  to  the  distance  of  the  earthy  being 
sojuetimes  a  little  gi-eater  and  at  other  times  a  little  less. 

When  the  apparent  diameter  of  the  moon  is  the  greater,  the 
shadow  will  extend  beyond  the  earth's  centre ;  and  when  the 
apparent  diameter  of  the  sun  is  the  greater,  it  will  fall  short  of  it. 
If  we  increase  the  mean  apparent  diameter  of  the  moon  as  seen 
from  the  earth's  centre,  viz :  31'  7 ",  by  gV)  the  ratio  of  the  radius 
of  the  earth  to  the  distance  of  the  moon,  we  shall  have  31'  38" 
for  the  mean  apparent  diameter  of  the  moon  as  seen  from  the 
nearest  point  of  the  earth's  surface.  Comparing  this  with  the 
mean  apparent  diameter  of  the  sun  as  viewed  from  the  same 
point,  which  is  sensibly  the  same  as  at  the  centre  of  the  earth,  or 
32'  2",  we  perceive  that  it  is  less  ;  from  which  we  conclude,  that 


PORTION  OF  earth's  SURFACE  WITHIN  THE  PENUMBRA.    173 

when  the  sun  and  moon  are  each  at  their  mean  distance  from  the 
earth,  the  shadow  of  the  moon  does  not  extend  as  far  to  the  earth's 
surface. 

434.  To  find  a  general  expression  for  the  length  of  the  moon's 
shadow,  let  A  G  B,  a'  g'  b',  and  a  gb  (Fig.  60).  be  sections  of  the 
sun,  moon,  and  earth,  by  a  plane  passing  through  their  centres 
S,  M,  and  E,  supposed  to  be  in  the  same  right  line,  and  A  a',  B 
b'  tangents  to  the  circles  A  G  B,  a'  g'  b  :  then  a'  K  b'  will  repre- 
sent the  moon's  shadow.  Let  L  —  the  length  of  the  shadow  ;  D 
=  the  distance  of  the  moon  ;  D'  =  the  distance  of  the  sun  ;  d  = 
apparent  semi-diameter  of  the  moon ;  and  o  =  apparent  semi- 
diameter  of  the  sun.  At  K  the  vertex  of  the  shadow,  M  K  a'  the 
apparent  semi-diameter  of  the  moon,  will  be  equal  to  S  K  A  the 
apparent  semi-diameter  of  the  sun ;  and  as  the  distance  of  this  point 
from  the  centre  of  the  earth,  even  when  it  is  the  greatest,  is  small 
in  comparison  with  the  distance  of  the  sun  (Art.  433),  the  apparent 
semi-diameter  of  the  sun  will  always  be  very  nearly  the  same  to  an 
observer  situated  at  K  as  to  one  situated  at  the  centre  of  the  earth. 
Now,  since  the  apparent  semi-diameter  of  the  moon  is  inversely 
proportional  to  its  distance, 

angle  M  K  a'  :  ^  :  :  M  E  :  M  K  ; 
and  thus,  8  :  d  : :  MB  :  MK  :  :  J)  :  L  (nearly) ; 

whence,  L  =  dA  .  .  .  (104). 

6 

If  a  more  accurate  result  be  desired,  we  have  only  to  repeat  the 

calculations,    after   having  diminished  o  in  the  ratio  of  D  to 

(D  -f  L). 

435.  Now,  to  find  the  breadth  of  the  portion  of  the  earth's  sur- 
face, covered  by  the  penumbral  shadow,  let  the  lines  A  d',  B  c' 
(Fig.  60)  be  drawn  tangent  to  the  circles  A  G  B,  «'  g'  b',  on 
opposite  sides,  and  prolonged  on  to  the  earth.  The  space  h  c'  d' 
k  will  represent  the  penumbra  of  the  moon's  shadow,  and  the  arc 
g  h  one  half  the  breadth  of  the  portion  of  the  earth's  surface 
covered  by  it.  Let  this  arc  or  the  angle  o-  E  A  =  S,  and  denote 
the  semi-diameter  of  the  sun,  and  the  semi-diameter  and  parallax 
of  the  moon,  by  the  same  letters  as  in  previous  Articles.  The 
triangle  M  E  A  gives, 

ande  MEA=S=MAZ— ^ME. 


174  ASTRONOMY. 

The  angle  h  M  E  is  the  idooii's  parallax  in  altitude  at  the  sta- 
tion A,  and  M  A  Z  is  its  zenith  distance  at  the  same  station.     De- 
note the  former  by  P'  and  the  latter  by  Z.     Thus, 
S  =  Z  ~  P'  .  .  .  (105). 

The  triangle  /t  M  S  gives, 

/iME  =P'  =  MS  A-f  MAS; 

M  A  S  =  cZ  -h  0  ;  and  M  S  h  is  the  sun's  parallax  in  altitude  at  the 
station  h  ;  let  it  be  denoted  by  ;j'.     We  have  then, 

P'  =  (/  +  0  +  p'  =  rZ  -i-  0  (nearly)  .  .  .  (106) ; 

and  to  find  Z  we  have  (equa.  11,  p.  43), 

P'  =  P  sin  Z,  or  sin  Z  =  ^-  .  .  .  (107). 

P'  and  Z  being  found  by  these  equations,  equa.  (105)  will  then 
make  known  the  value  of  S. 

If  great  accuracy  is  required,  the  calculation  must  be  repeated, 
giving  now  to  j)'  in  equation  (106)  the  value  furnished  by  equa- 
tion (11)  which  expresses  the  relation  between  the  parallax  in 
altitude  of  a  body  and  its  horizontal  parallax,  instead  of  ne- 
glecting it  as  before ;  and  Z  must  be  computed  from  the  follow- 
ing equation  : 

sinZ=^H?^'  .  .  .  (108). 
sm  P  ^       ^ 

The  penumbral  shadow  will  obviously  attain  to  its  greatest 
breadth,  when  the  sun  is  in  its  perigee  and  the  moon  is  in  its 
apogee.  The  values  oi  d  6  and  P  under  these  circumstances 
are  respectively  14'  41",  16'  18",  and  53'  48".  Performing  the 
calculations,  we  find  that  the  breadth  of  the  greatest  portion  of 
the  earth's  surface  ever  covered  by  the  penumbral  shadoiv,  is 
69°  17',  or  about  4800  miles. 

436.  The  breadth  of  the  spot  comprehended  within  the  um- 
bra, may  be  found  in  a  similar  manner.  The  arc  g  h'  (Fig.  60) 
represents  one  half  of  it :  denote  this  arc  or  the  equal  angle  g  E 
h'  by  S', 

M  E  A'  =  S'  =  M  /i'  Z'  —  A'  M  E  ; 
or,  S'  =  Z  — P'  .  .  .  (109). 

A'ME  =  P'  =  MSA'  +  MA'S; 


PORTION    OF    F:aRTH'S    SIRFACE    WITHIN    THE    UMBRA.       175 

but  M  h'  S  =  d  —  (5,  and  M  S  h'  =  />',  sun's  parallax  in  altitude  at 
h'  ]  whence, 

V  =  d  —  8  -\-p'  =^d  —  S  {nesLYlY)  •  .  •  (110); 
and  we  have,  as  before, 

P'  =  P  sm  Z,  or  sin  Z  =  !^  .  .  .  (111). 

The  greatest  breadth  will  obtain  when  the  sun  is  in  its  apo- 
gee and  the  moon  is  in  its  perigee.     We  shall  then  have, 
6  =  15'  45 ",  d  =  16'  45 ",  P  =  61'  24". 

Making  use  of  these  numbers,  we  deduce  for  the  rnaximuni 
breadth  of  the  portion  of  the  earths  surface  covered  hy  the 
moon^s  shadow,  1°  50',  or  127  miles. 

437.  It  should  be  observed  that  the  deductions  of  the  last  two 
articles  answer  to  the  supposition  that  the  moon  is  in  the  node, 
and  that  the  axis  of  the  shadow  and  penumbra  passes  through 
the  centre  of  the  earth.  In  every  other  case,  both  the  shadow 
and  penumbra  will  be  cut  obliquely  by  the  earth's  surface,  and 
the  sections  will  be  ovals,  and  very  nearly  true  ellipses,  the 
lengths  of  which  may  materially  exceed  the  above  determi- 
nations. 

438.  Parallax  not  only  causes  the  eclipse  to  be  visible  at 
some  places  and  invisible  at  others,  as  shown  in  Art.  431 ;  but, 
by  making  the  distance  of  the  centres  of  the  sun  and  moon  un- 
equal, renders  the  circumstances  of  the  eclipse  at  those  places 
where  it  is  visible,  different  at  each  place.  This  may  also  be 
inferred  from  the  circumstance  that  the  different  places,  covered 
at  any  time  by  the  shadow  of  the  moon,  v.'ill  be  differently  situ- 
ated within  this  shadow.  It  will  be  seen,  therefore,  that  an 
eclipse  of  the  sun  has  to  be  considered  in  two  points  of  view : 
1st.  With  respect  to  the  lohole  earth,  or  as  a  general  eclipse; 
and  2d.    With  respect  to  a  jjarticular  place. 

439.  The  following  are  the  principal  facts  relative  to  eclipses 
of  the  sun  that  remain  to  be  noticed.  1st.  The  duration  of  a 
general  eclipse  of  the  sun  cannot  exceed  about  6  hours.  2d.  A 
solar  eclipse  does  not  happen  at  the  same  time  at  all  places 
where  it  is  seen  ;  as  the  motion  of  the  moon  beyond  the  sun,  and 
consequently  of  its  shadow,  is  from  west  to  east,  the  eclipse 
must  begin  earlier  at  the  western  parts  and  later  at  the  eastern. 


17G  ASTRONOMY. 

3d.  The  moon's  shadow  beins:  tangent  to  the  earth,  at  the  com- 
mencement and  end  of  the  eclipse,  the  sun  will  be  just  rising  at 
the  place  where  the  eclipse  is  first  seen,  and  just  setting  at  the 
place  where  it  is  last  seen.  At  the  intermediate  places,  the  sun 
will  at  the  time  of  the  beginning  and  end  of  the  eclipse  have 
various  altitudes.  4th.  An  eclipse  of  the  snn  begins  on  the 
western  side  and  ends  on  the  eastern.  5th.  When  the  straight 
line  passing  through  the  centres  of  the  sun  and  moon,  passes 
also  through  the  place  of  the  spectator,  the  eclipse  is  said  to  be 
central:  a  central  eclipse  may  be  either  annular  or  total,  ac- 
cordinsT  as  the  apparent  diameter  of  the  sun  is  greater  than  that 
of  the  moon,  or  the  reverse.  6th.  A  total  eclipse  of  the  sun  can- 
not last  at  any  one  place  more  than  eight  minutes  ;  and  an  an- 
nular eclipse  more  than  twelve  and  a  half  minutes.  7th.  In 
most  solar  eclipses  the  moon's  disc  is  covered  with  a  faint  light, 
a  phenomenon  which  is  attributed  to  the  reflection  of  the  light 
from  the  illuminated  part  of  the  earth. 

Calculation  of  an  Eclipse  of  the  Sun. 
1.  Of  the  circumstances  of  the  general  eclipse. 

440.  It  is  a  simple  inference  from  what  has  been  established 
in  Art.  426,  that  an  eclipse  of  the  sun  will  begin  and  end  upon 
the  earth,  at  the  times  before  and  after  conjunction,  when  the 
distance  of  the  centres  of  the  moon  and  sun  is  equal  to  P  —  p  -f 
^  +  d\  that  the  total  eclipse  will  begin  and  end  when  this  dis- 
tance is  equal  toP  — p  —  5-f-  d  ]  and  the  annular  eclipse  when 
the  distance  is  equal  to  P  —  p  +  5  —  d. 

441.  The  times  of  the  various  phases  of  the  general  eclipse  of 
the  sun,  may  be  obtained  by  a  process  precisely  analogous  to 
that  by  which  the  times  of  those  of  an  eclipse  of  the  moon  are 
found.  Let  C  (Fig.  61)  be  the  centre  of  the  sun,  and  C 
the  centre  of  the  moon,  at  the  time  of  conjunction.  We  may 
suppose  the  sun  to  remain  stationary  at  C,  if  we  attribute  to  the 
moon  a  motion  equal  to  its  motion  relative  to  the  sun  ;  for,  on 
this  supposition,  the  distance  of  the  centres  of  the  two  bodies 
Avill,  at  any  given  period  during  the  eclipse,  be  the  same  as  that 
which  obtains  in  the  actual  state  of  the  case.  Let  N'  C  L'  re- 
present the  orbit  that  would  be  described  by  the  moon  if  it  had 
such  a  motion,  which  is  called  the  Relative  Orbit.  Let  C  M  be 
drawn  perpendicular  to  it ;  and  let  C/=  C/'  =  P  —  p  +  6  +  d, 


SOLAR    ECLIPSE — CIRCUMSTANCES    OP  GENERAL    ECLIPSE.    177 

and  C^  =  C^'  =  P  —  p  —  o+  c?,  orP  —  p  +  6  —  d,  according- 
as  the  eclipse  is  total  or  annular.  Then,  M  will  be  the  place  of 
the  moon's  centre  at  the  middle  of  the  eclipse  ; /and/' the 
places  at  the  beginning  and  end  of  the  eclipse ;  and  g  and  g' 
the  places  at  the  beginning  and  end  of  the  total,  or  of  the  annu- 
lar eclipse.     We  shall  thus  have,  as  in  eclipses  of  the  moon, 

tang  I  =  -JL ,  C  M  =X  cos  I,  C  M  =  X  sin  I .  .  .  (112). 

^  M  —  m  ^       ' 

T  .        ir                *        J        3600s.  X  sin  I  COS  I  /i-(o\ 

Interval  from  con.  to  mid.  — .  .  .  (113). 

M  —  m.  ^       ' 

Interval  from  middle  to  beginning  or  end 
3600s  cos  I 


M  —  m 


v/(yt' +  XcosI)(A;'  — XcosI)  •  •  •  (114). 


Interval  for  total  eclipse 

3600s.  cos  I /I -IPX 

=  -^— -^^  sf  {k"  +  X  cos  I)  {k"  —  X  cos  I)  •   .  •  (115). 

Interval  for  annular  eclipse 

3600s.  cos  I    , /i-./'x 

=  —^_zr^ —  ^(^-"'  +  >-  cos  I)  (^•"'  —  X  cos  I)  .  .  •  (116). 

Quantity  =.5i^-:Z?l.^£^)  .  .  .  (117). 

(ju 

A-' =  P_p4-5+(/,  A:"=  P— p  —  5-frf,  A'"' =  P— p +'<5  — </ (118), 

The  letters  X,  M,  w,  &.c.  represent  quantities  of  the  same  name 
as  in  the  formulae  for  a  lunar  eclipse;  but  they  designate  the 
values  of  these  quantities  at  the  time  of  conjunction.,  instead  of 
opposition.  These  values  are  in  practice  obtained  from  tables 
of  the  sun  and  moon,  as  in  a  lunar  eclipse. 

442.  The  times  of  the  different  circumstances  of  a  general 
eclipse  of  the  sun,  may  also  be  found  within  a  minute  or  two  of 
the  truth,  by  construction,  in  a  precisely  similar  manner  with 
those  of  an  eclipse  of  the  moon  (Art.  423). 

2.  Of  the  phases  of  the  eclipse  at  a  particular  place. 

443.  The  phase  of  the  eclipse,  which  obtains  at  any  instant 
at  a  given  place,  is  indicated  by  the  relation  between  the  appa- 
rent distance  of  the  centres  of  the  sun  and  moon  and  the  sum  of 
their  apparent  semi-diameters.     And  the  calculation  of  the  time 

23 


178  ASTRONOMY. 

of  any  given  phase  of  the  eclipse,  consists  in  the  calculation  of 
the  time  when  the  apparent  distance  of  the  centres  has  the  value 
relative  to  the  sum  of  the  semi-diameters,  answering  to  the  given 
phase.  Thus,  if  we  wish  to  find  the  time  of  the  beginning  of 
the  eclipse,  we  have  to  seek  the  time  when  the  apparent  dis- 
tance of  the  centres  of  the  sun  and  moon  first  becomes  equal  to 
the  sum  of  their  apparent  semi-diameters. 

444.  The  calculation  of  the  different  phases  of  an  eclipse  of 
the  sun,  for  a  particular  place,  involves,  then,  the  determination 
of  the  apparent  distance  of  the  centres  of  the  sun  and  moon,  and 
of  the  apparent  semi-diameters  of  the  two  bodies  at  certain  stated 
periods. 

The  true  semi-diameter  of  the  sun,  as  given  by  the  tables, 
may  be  taken  for  the  apparent  without  material  error.  For  the 
method  of  computing  the  apparent  semi-diameter  of  the  moon, 
for  any  given  time  and  place,  see  Problem  XXVII. 

445.  According  to  the  celebrated  astronomer  Dusejour,  in  or- 
der to  make  the  observations  agree  with  theory,  it  is  necessary 
to  diminish  the  sun's  semi-diameter,  as  it  is  given  by  the  tables, 
3". 5.  This  circumstance  is  explained,  by  supposing  that  the 
apparent  diameter  of  the  sun  is  amplified,  by  reason  of  the  very 
lively  impression  which  its  light  makes  upon  the  eye.  This 
amplification  is  called  Irradiation.  He  also  thinks  that  the 
semi-diameter  of  the  moon  ought  to  be  diminished  2",  to  make 
allowance  for  an  Inflexion  of  the  light  which  passes  near  the 
border  of  this  luminary,  supposed  to  be  produced  by  its  atmos- 
phere. It  must  be  observed,  however,  that  the  astronomers  of 
the  present  day  do  not  agree  either  as  to  the  necessity  or  the 
amount  of  the  diminutions  just  spoken  of. 

446.  The  determination  of  the  apparent  distance  of  the  cen- 
tres of  the  sun  and  moon  may  easily  be  accomplished,  as  will 
be  shown  in  the  sequel,  when  the  apparent  longitude  and  lati- 
tude of  the  two  bodies  have  been  found.  Now,  the  true  longi- 
tude of  the  sun,  and  the  true  longitude  and  latitude  of  the  moon, 
may  be  found  from  the  tables  (Probs.  IX  and  XIV)  ;  and 
from  these  the  apparent  longitudes  and  latitudes  may  be  de- 
duced, by  correcting  for  the  parallax.  But  the  customary 
mode  of  proceeding  is  a  little  different  from  this  :  the  true  longi- 
tude and  latitude  of  the  sua  are  employed  instead  of  the  appa- 


SOLAR    ECLIPSE. APPROXIMATE    TIMES    OP    PHASES.  179 

rent,  and  the  parallax  of  the  sun  is  referred  to  the  moon  :  that  is, 
the  difference  between  the  parallax  of  the  moon  and  that  of  the 
sun  is,  by  fiction,  taken  as  the  parallax  of  the  moon.  This  sup- 
posititious parallax  is  called  the  moon's  Relative  Parallax. 
(See  Prob.  XVII.) 

447.  We  will  first  show  how  to  find  the  approximate  times 
of  the  different  phases  of  the  eclipse.  Put  T  =  the  time  of 
new  moon  known  to  within  5  or  10  minutes.  (Prob.  XXVII.) 
For  the  time  T  calculate  by  the  tables  the  sun's  longitude, 
hourly  motion,  and  semi-diameter,  and  the  moon's  longitude, 
latitude,  horizontal  parallax,  semi-diameter,  and  hourly  motions 
in  longitude  and  latitude.  Subtract  the  sun's  horizontal  paral- 
lax from  the  reduced  horizontal  parallax  of  the  moon,*  and  cal- 
culate the  apparent  longitude  and  latitude,  and  the  apparent 
semi-diameter  of  the  moon.  From  a  comparison  of  the  apparent 
longitude  of  the  moon  with  the  true  longitude  of  the  sun,  we 
shall  know  whether  apparent  ecliptic  conjunction  occurs  before 
or  after  the  time  T.  Let  T'  denote  the  time  an  hour  earlier  or 
later  than  the  time  T,  according  as  the  apparent  conjunction  is 
earlier  or  later.  With  the  sun  and  moon's  longitudes,  the 
moon's  latitude,  and  the  hourly  motions  in  longitude  and  lati- 
tude at  the  time  T,  calculate  the  longitudes  and  the  moon's  lati- 
tude for  the  time  T' ;  and  for  this  time  also  calculate  the  moon's 
apparent  longitude  and  latitude.  Take  the  difference  between 
the  apparent  longitude  of  the  moon  and  the  true  lono^itude  of 
the  sun  at  the  time  T,  and  it  will  be  the  apparent  distance  of 
the  moon  from  the  sun  in  longitude,  at  this  time.  Let  it  be  de- 
noted by  n.  Find,  in  like  manner,  the  apparent  distance  of  the 
moon  from  the  sun  in  longitude  at  the  time  T',  and  denote  it  by 
n'.  In  the  same  manner  as  at  the  time  T,  we  find  whether  ap- 
parent conjunction  occurs  before  or  after  the  time  T'.  If  it  oc- 
curs between  the  times  T  and  T',  the  sum  of  n  and  n',  otherwise 
their  difference,  will  be  the  apparent  relative  motion  of  the  sun 
and  moon  in  longitude  in  the  interval  T'  —  T,  or  T  —  T'  ;  from 
which  the  relative  hourly  motion  will  become  known.  The  dif- 
ference of  the  apparent  latitudes  of  the  moon,  at  the  times  T  and 

*  The  reduced  horizontal  parallax  of  the  moon  is  its  horizontal  parallax  as  re- 
duced from  the  equator  to  the  given  place.     (See  Prob.  XV.) 


180  ASTRONOMY. 

T',  will  make  known  the  apparent  relative  hourly  motion  in  lati- 
tude.    With  the  relative  hourly  motion  in  longitude  and  the  dif- 
ference of  the  apparent  longitudes  at  the  time  T,  find  by  simple 
proportion,  the  interval  between  the  time  T  and  the  time  of  ap- 
parent ecliptic  conjunction  :  and  then,  with  the  apparent  latitude 
of  the  moon  at  the  time  T  and  its  hourly  motion  in  latitude, 
find  the  apparent  latitude  at  the  time  of  apparent  conjunction 
thus  determined.     Then,  knowing  the  relative  hourly  motion 
of  the  sun  and  moon  in  longitude  and  latitude,  together  with 
the  time  of  apparent  conjunction,  and  the  apparent  latitude  at 
that  time,  and  regarding  the  apparent  relative  orbit  of  the  moon 
as  a  right  line,  (which  it  is  nearly,)  it  is  plain  that  the  time  of 
beginninsT,  greatest  obscuration,  and  end,  as  well  as  the  quantity 
of  the  eclipse,  may  be  calculated  after  the  same  manner  as  in 
the  general  eclipse  ;  the  disc  of  the  sun  answering  to  the  sec- 
tion of  the  luminous  frustum  mentioned  in  Art.  424,  and  the 
apparent  elements  answering  to  the  true.     Let  C  (Fig.  62)  repre- 
sent the  centre  of  the  sun  supposed  stationary,  C  C  the  apparent 
latitude  of  the  moon  at  apparent  conjunction,  N'  C  the  apparent 
relative  orbit  of  the  moon,  determined  by  its  passing  through  the 
point  C,  and  making  a  determinate  angle  with  the  ecliptic  N'  F, 
or  by  its  passing  through  the  situations  of  the  moon  at  the  times 
T  and  T'.     Also,  let  R  K  F  K'  be  the  border  of  the  sun's  disc  ; 
/,/'  the  positions  of  the  moon's  centre  at  the  beginning  and  end 
of  the  eclipse,  determined  by  describing  a  circle  around  C  as  a 
centre,  with  a  radius  equal  to  the  sum  of  the  apparent  semi- 
diameters  of  the  sun  and  moon  ;  and  M  (the  foot  of  the  perpen- 
dicular let  fall  from  0  upon  N'  C)  its  position  at  the  time  of 
greatest  obscuration. 

If  the  eclipse  should  be  total  or  annular,  then  g,  g'  will  be  the 
positions  of  the  moon's  centre  at  the  beginning  and  end  of  the 
total  or  annular  eclipse ;  these  points  being  determined  by 
describing  a  circle  around  C  as  a  centre,  and  with  a  radius  equal 
to  the  difierence  of  the  apparent  semi-diameters  of  the  sun  and 
moon. 

The  various  circumstances  of  the  eclipse  may  also  be  had  by 
construction,  after  the  same  manner  as  in  a  lunar  eclipse  (Art. 
423). 

448.  In  order  to  be  able  to  observe  the  beginning  or  end  of  a 


SOLAR   ECLIPSE POINTS    OF    CONTACT.  181 

solar  eclipse,  it  is  necessary  to  know  the  position  of  the  point  on 
the  sun's  limb  where  the  first  or  last  contact  takes  place.  The  sit- 
uation of  these  points  is  designated  by  the  distance  on  the  limb, 
intercepted  between  them  and  the  highest  point  of  the  limb,  called 
the  Vertex.  The  contacts  will  take  place  at  the  points  /,  t'  (Fig. 
62),  on  the  lines  C/,  C/'.  To  find  the  position  of  the  vertex, 
with  the  sun's  longitude  found  for  the  beginning  of  the  eclipse 
calculate  the  angle  of  position  of  the  sun  at  that  time,*  and 
lay  it  off  to  the  right  of  the  circle  of  latitude  C  K  when  the  sun's 
longitude  is  between  90°  and  270°,  and  to  the  left  when  the  longi- 
tude is  less  than  90°  or  more  than  270°.  Suppose  C  P  to  be  the 
circle  of  declination  thus  determined.  Next,  let  Z  (Fig.  18)  be 
the  zenith,  P  the  elevated  pole,  and  S  the  sun  ;  then  in  the  triangle 
Z  P  S  we  shall  know  Z  P  the  co-latitude,  Z  P  S  the  hour  angle 
of  the  sun,  and  we  may  deduce  P  S  the  co-declination  of  the  sun, 
from  the  longitude  of  the  sun  as  derived  from  the  tables,  (equa. 
36).  These  three  quantities  being  known,  Z  S  P  the  angle  made 
by  the  vertical  through  the  sun  with  its  circle  of  declination,  may 
be  computed  ;  and  being  laid  off"  in  the  figure  to  the  right  or  left 
of  C  P  (Fig.  62),  according  as  the  time  of  beginning  is  before  or 
after  noon,  the  point  Z  or  Z',  as  the  case  may  be,  in  which  the 
vertical  intersects  the  limb  R  K  K',  will  be  the  vertex,  and  the 
arc  Z  /,  or  Z'  t  on  the  limb,  will  ascertain  the  situation  of  t  the 
first  point  of  contact,  with  respect  to  it. 

The  situation  of  the  last  point  of  contact  may  be  found  by  the 
same  mode  of  proceeding. 

449.  Let  us  now  show  how  to  find  the  exact  times  of  the  begin- 
ning, greatest  obscuration,  and  end  of  the  eclipse,  the  approxi- 
mate times  being  known.  Let  B  designate  the  approximate  time 
of  beginning,  taken  to  the  nearest  minute.  Calculate  for  the  time 
B  by  means  of  the  tables,  the  sun's  longitude,  hourly  motion,  and 
semi-diameter ;  also  the  moon's  longitude,  latitude,  horizontal 
parallax,  semi-diameter,  and  hourly  motions  in  longitude  and 
latitude.  Then,  making  use  of  the  relative  parallax,  calculate 
the  apparent  longitude,  latitude,  and  semi-diameter  of  the  moon. 


*  A  formula  which  makes  known  the  angle  of  position  of  the  sun,  when  the  Ion- 
gitude  of  the  sun  and  the  obliquity  of  the  ecliptic  are  given,  is  investigated  in  the 
Appendix.     (See  Prob.  XIII). 


182  ASTRONOMY. 

Subtract  the  apparent  longitude  of  the  moon  from  the  true  longi- 
tude of  tlie  sun  ;  the  difference  will  be  the  apparent  distance  of 
the  moon  from  the  sun  in  longitude  ;  let  it  be  denoted  by  a.  De- 
note the  apparent  latitude  of  the  moon  by  X. 

Now,  let  E  C  (Fig.  63)  represent  an  arc  of  the  ecliptic,  and  K 
its  pole  ;  and  let  S  be  the  situation  of  the  sun,  and  M  the  apparent 
situation  of  the  moon  at  the  time  B.  Then  M  S  is  the  apparent 
distance  of  the  centres  of  the  two  bodies  at  this  time.  Denote  it 
by  A.  S  m,=  a,  and  M  w  =  X.  The  right  angled  triangle  M  S 
ni  being  very  small,  may  be  considered  as  a  plane  triangle,  and 
we  therefore  have,  to  determine  A,  the  equation, 

A^   =«-    +   X2    ...    (119).* 

450.  Having  computed  the  value  of  a,  we  find,  by  comparing 
it  with  the  sum  of  the  apparent  serai-diameters  of  the  sun  and 
moon,  whether  the  beginning  of  the  eclipse  occurs  before  or  after 
the  approximate  time  B.  Fix  upon  a  time  some  4  or  5  minutes 
before  or  after  B,  according  as  the  beginning  is  before  or  after,  and 
call  it  B'.  With  the  sun  and  moon's  longitudes,  the  moon's  lati- 
tude, and  the  hourly  motions  in  longitude  and  latitude,  at  the  time 
B,  find  the  longitudes  and  the  moon's  latitude  at  the  time  B',  and 
compute  for  this  time  the  apparent  longitude,  latitude,  and  semi- 
diameter  of  the  moon.  Subtract  the  apparent  longitude  of  the  moon 
from  the  true  longitude  of  the  sun,  and  we  shall  have  the  apparent 
distance  of  the  moon  from  the  sun  at  the  time  B'.  Subtract  this 
from  the  same  distance  a  at  the  time  B,  and  we  shall  have  the  appa- 
rent relative  motion  of  the  sun  and  moon  in  longitude  durins:  the 
interval  of  time  between  B  and  B'.  Then  find,  by  simple  pro- 
portion, the  apparent  relative  hourly  motion  in  longitude,  and 
denote  it  by  k.  Take  the  dilference  between  the  apparent  lati- 
tudes of  the  moon  at  the  times  B  and  B',  and  it  will  be  the  appa- 
rent relative  motion  of  the  sun  and  moon  in  latitude,  in  the  inter- 
val ;  from  which  deduce  the  apparent  relative  hourly  motion  in 
latitude,  and  call  it  n.     Now,  put  t  =  the  interval  between  the 


*  In  place  of  equation  (119),  the  following  equations  may  be  employed  in  loga- 
rithmic computation  : 


where  6  is  an  auxiliary  arc. 


tang  9  =  _ ,  A  = 

a  cos  9 


SOLAR    ECLIPSE TRUE    TIMES    OF    PHASES.  183 

approximate  and  true  times  of  the  beginning  of  the  eclipse,  and 
suppose  S  and  M  (Fig.  63)  to  be  the  situations  of  the  sun  and 
moon  at  the  true  time  of  beginning.  In  the  time  /,  the  apparent 
relative  motions  in  longitude  and  latitude,  will  be,  respectively, 
equal  to  k  t  and  n  t,  and  accordingly  we  shall  have, 
S  m  =  a  —  k  t,  M  ?;i  =  X  +  n  t. 
The  small  right-angled  triangle  S  M  in  may  be  considered  as  a 
plane  triangle  ;  the  hypothenuse  S  M  =  4'  =  the  sum  of  the  appa- 
rent semi-diameters  of  the  sun  and  moon,  minus  5". 5.  We  have 
then,  to  find  t,  the  equation, 

{a—  k  ty-  +  (X  +  n  ty-  =  ^% 

or,  developing  and  transposing, 

{n-  +  k^-)r-  ^2{ak  —  Xti)  #  =  4-2  _(«=  +  X2)  =  4.3  —  A3  ; 
making  A  =  4^-  —  A-,  and  B  —  a  k  —  X  7^, 
[n^-  +  ^•=)  r-  ~2Bt  =  A, 


and,  t  ^ B-VB^  +  AJ^^^^)  (^^^^ 

n^  +  k^ 
The  negative  sign  must  be  prefixed  to  the  radical,  for,  if  we 
suppose  A  to  be  equal  to  zero,  t  must  be  equal  to  zero.     Multiply- 
ing the  numerator  and  denominator  by  B  +  ■/  B-  +  A(n^  +  k"), 
and  restoring  the  value  of  A,  we  obtain, 

3600s.  (4.2  —  A2), 
(in  seconds)  t  = 7^=^-,-^—    ,  ,  /  — =^^  •  •  (121). 

Although  this  equation  has  been  investigated  for  the  begin- 
ning of  the  eclipse,  it  is  plain  that  it  will  answer  equally  well  for 
the  determination  of  the  other  phases,  if  we  give  the  proper  val- 
ues and  signs  to  4^,  «,  "^i  n,  and  k.  k  is  positive  before  conjunc- 
tion and  negative  after  it ;  ti  is  negative,  when  the  moon  appears 
to  recede  from  the  north  pole  of  the  ecliptic  ;  X  is  negative,  when 
it  is  south  ;  a  is  always  positive.* 


*  Developing  the  radical  in  equation  (120),  and  neglecting  all  the  terms  after 
the  second,  as  being  very  small,  we  obtain  tor  the  beginning  and  end  of  the  eclipse 
the  more  convenient  formula, 

^  _  1800s.  jxp"^  —  A^) 
B 


184  ASTRONOMY. 

451.  The  values  of  the  quantities  a  X,n,  and  k,  are  found  for 
the  other  phases  after  the  same  manner  as  for  the  beginning. 

To  obtain  the  vahie  of  -^  at  the  time  of  greatf  st  obscuration, 
find  the  relative  motions  in  longitude  and  latitude  {k  and  n), 
during  some  short  interval  near  the  middle  of  the  eclipse,  which 
is  the  approximate  time  of  greatest  obscuration  ;  then  compute 
the  inclination  of  the  relative  orbit  by  the  equation 

tang  l=—  .  .  .  (122).  (See  equa.  90) : 
k 

after  which  -^  will  result  from  the  equation 

4.  =  X  cos  I  .  .  .  (123).     (See  equa.  94). 

For  the  beginning  and  end  of  the  total  eclipse,  we  have,  -^  = 

appar.  semi-diam.  of  moon  —  appar.  semi-diam.  of  sun  +  1".5 ; 

and  for  the  beginning  and  end  of  the  annular  eclipse,  4^  =  appar. 

semi-diam.  of  sun,  —  appar.  semi-diam.  of  moon,  —  1  ".5. 

452.  If  the  value  of  4^,  given  by  equation  (123),  be  substituted 
in  equation  (121),  this  equation  will  make  known  the  time  of 
greatest  obscuration  ;  but  this  may  be  found  more  conveniently 
by  a  different  process.  Let  N  C  F  (Fig.  64)  represent  a  por- 
tion of  the  ecliptic,  E  M  L  a  portion  of  the  relative  orbit 
passed  over  about  the  time  of  greatest  obscuration,  C  the  sta- 
tionary position  of  the  sun's  centre,  and  M  the  place  of  the 
moon's  centre  at  the  instant  of  its  nearest  approach  to  C.  Also, 
let  a  =  C  R  the  apparent  distance  of  the  moon  from  the  sun  in 
longitude  at  the  time  of  the  nearest  approach  of  the  centres, 
X'  =  R  M  the  moon's  apparent  latitude  at  the  same  time,  k  =  M  k 
the  apparent  relative  motion  in  longitude  in  some  short  interval 
about  this  time,  and  n  =  k  n  the  moon's  apparent  motion  in  lati- 
tude during  the  same  interval.  The  right  angled  triangles  M 
71  k  and  C  M  R  are  similar,  for  their  sides  are  respectively  per- 
pendicular to  each  other  ;  whence, 

mk:MR::kn:CR] 

and  CR  =  MR  A!i,or,a  =  X'?  .  .  .  (124). 

Mk  k  ^       ^ 

If  the  moon's  apparent  latitude  be  found  for  the  approximate 
time  of  greatest  obscuration,  and  substituted  for  X'  in  equation 
(124),  this  equation  will  give  very  nearly  the  apparent  distance 


SOLAlt    ECLIPSE — TR,UE    TIIIES    OP    PHASES.  185 

(a)  of  the  two  bodies  in  longitude  at  the  true  time  of  greatest 
obscuration.  Witli  this,  and  the  apparent  distance  at  the  ap- 
proximate time  of  greatest  obscuration,  together  with  the  rela- 
tive apparent  motion  in  longitude,  the  true  time  of  greatest  ob- 
scuration may  be  found  nearly  by  simple  proportion.  A  more 
accurate  result  may  then  be  liad  by  finding  the  moon's  apparent 
latitude  for  the  time  obtained,  substituting  it  for  X'  in  equation 
(124)  and  then  repeating  the  calculations. 

453.  A  simpler,  though  less  accurate  method  than  that  already 
given,  of  finding  the  times  of  beginning  and  end  of  the  total  or 
annular  eclipse,  is  to  compute  the  half  duration  of  the  total  or 
annular  eclipse,  and  add  it  to,  and  subtract  it  from,  the  time  of 
greatest  obscuration.  This  interval  may  easily  be  determined, 
if  we  can  find  the  rate  of  motion  on  the  relative  orbit,  and  the 
distance  passed  over  by  the  moon's  centre  during  the  interval. 
Let  g,  g'  (Fig.  64)  be  the  places  of  the  moon's  centre  at  the  in- 
stants of  the  two  interior  contacts,  and  M  m  the  distance  passed 
over  in  some  short  interval  (L).  Let  ^  =  Z.  M  n  A*  the  comple- 
ment of  the  inclination  of  the  relative  orbit,  k  =  M  k,  k'  = 
M  >i,  and  t  =  half  duration  of  total  or  annular  eclipse.  The  tri- 
angles M  11  k,  C  R  M  give, 

Mn^   .  ^lX-,0Tk'  =  J^.  .  .  (125), 
sm  M  n  k  sm  d 

and,      tanor RC M  =  tang Wnk^ ^^,  or,  tang  6  =  —.  .  (126). 

C  R  a 

Findino;  the  value  of  d  by  the  last  equation  and,  si:'::r'tvting  it 
ill  aquation  (125),  we  obtain  the  value  of/;'  ;  and  then,  to  find  ty 
we  have, 

JIT  HT  ^  ^  L   X  M  ^ 

A:'  :  L  : :  M  o-  :  /,  or  ^  = 77— ^• 


M^=  ^C^'  — CM' =  %/4.2  —  A^    (Art.  451): 


L    v'^^— A^_L        (^+A)(4.— A) 

whence,  ^  = 77 j:, •  •  •  (l^i^j. 

454.  The  apparent  distance  of  the  centres  of  the  two  bodies 
at  the  time  of  greatest  obscuration  being  known,  the  quantity  of 
the  eclipse  may  be  readily  found.  We  have  but  to  subtract  the 
apparent  distance  from  the  sum  of  the  apparent  semi-diameters, 

24 


186  ASTRONOMY. 

and  state  the  proportion,  as  the  sun's  apparent  diameter  :  the  re- 
mainder :  :  12  digits  :  the  digits  eclipsed.  (For  a  more  particu- 
lar description  of  the  method  of  calculating  a  solar  eclipse,  see 
Prob.  XXX). 

Occultations. 

455.  At  all  places  upon  the  earth's  surface,  which  at  a  given 
time  have  the  moon  in  the  horizon,  its  apparent  place  will  differ 
from  its  true  place,  by  the  amount  of  its  horizontal  parallax.  It 
follows,  therefore,  that  a  star  will  be  eclipsed  by  the  moon  some- 
where upon  the  earth,  in  case  its  true  distance  from  the  moon's 
centre  is  less  than  the  sum  of  the  moon's  semi-diameter  and 
horizontal  parallax. 

The  greatest  value  of  the  moon's  semi-diameter  is  16'  45", 
and  that  of  its  horizontal  parallax  61'  24".  If  we  add  the  sum 
of  these  numbers  to  5°  17'  34",  the  maximum  latitude  of  the 
moon,  we  obtain  as  the  result  6°  35'  43".  It  is  then  only  the 
stars  which  have  a  latitude  less  than  6^  35'  43",  that  can  ex- 
perience an  occultation  from  the  moon. 

456.  By  considering  the  various  situations  of  the  stars  liable 
to  an  occultation,  taking  the  greatest  and  least  values  of  the  sum 
of  the  moon's  semi-diameter  and  horizontal  parallax,  and  allow- 
ing for  the  inequalities  of  the  motions  of  the  moon,  it  has  been 
found,  that,  if  at  the  time  of  the  mean  conjunction  of  the  moon 
and  a  star,  (that  is,  when  the  moon's  mean  longitude  is  the  same 
with  the  longitude  of  the  star,)  their  difference  of  latitude  ex- 
ceed 1°  37',  there  cannot  be  an  occultation  ;  if  the  difference  be 
less  than  51',  there  must  be  an  occultation  somewhere  on  the 
earth  ;  and  that  between  these  limits  there  is  a  doubt,  which 
can  only  be  removed  by  the  calculation  of  the  moon's  true 
place. 

457.  The  calculation  of  an  occultation  is  very  nearly  the 
same  as  that  of  a  solar  eclipse.  The  only  difference  is  in  the 
data.  The  star  has  no  diameter,  parallax,  or  motion  in  longi- 
tude ;  and,  as  it  is  situated  without  the  ecliptic,  we  have,  in 
place  of  the  latitude  of  the  moon,  employed  in  solar  eclipses, 
the  difference  between  the  latitude  of  the  moon  and  that  of  the 
star,  and  in  place  of  the  difference  between  the  longitudes  of 
the  two  bodies  and  their  relative  hourly  motion  in  longitude, 
these  quantities  referred  to  an  arc  passing  through  the  star  and 


OCCULTATIONS.  187 

parallel  to  the  ecliptic.  Thus,  if  E  C  (Fig.  63)  represent  the 
ecliptic,  K  its  pole,  s  the  situation  of  the  star,  M  that  of  the 
moon,  and  s  ml  an  arc  passing  through  s  and  parallel  to  the 
arc  E  C,  we  have  in  place  of  ^/z,  M  ,  m'  M  =  wi  M  —  7n  in\  and 
in  place  of  S  m,  s  m'.  The  hourly  variation  of  S'  m  must  also 
be  reduced  to  the  arc  5  in'. 

458.  The  reduction  of  the  difference  of  longitude  of  the 
moon  and  star,  to  the  parallel  to  the  ecliptic,  passing  through  the 
star,  is  effected  by  multiplying  this  difference  by  the  cosine  of 
the  latitude  of  the  star.  For,  let  A  B  (Fig.  65)  be  an  arc  of  the 
ecliptic,  and  A'  B'  the  corresponding  arc  of  a  circle  parallel  to  it ; 
then,  since  similar  arcs  of  circles  are  proportional  to  their  radii, 
we  have, 

BC  :B'C':  :  AB:  A'B'  =^^-^^ 


BC 
but,  B'C'  =  Ca-B'C  cosBCB'  =  BC  cos  BB'. 

Hence,        A'  B'  =  AB.BCcos^B;^  =  A  B  cos  B  B'. 

BC 

The  reduction  of  the  relative  hourly  motion  in  longitude  to 
the  parallel  in  question,  is  obviously  effected  in  the  same 
manner. 


CHAPTER    XVII. 

OF    THE    PLANETS    AND    THE    PHENOMENA    OCCASIONED    BY 
THEIR    MOTIONS    IN    SPACE. 

Apparent  Motions  of  the  Planets  with  respect  to  the  Sun. 

459.  The  apparent  motion  of  an  inferior  planet,  with  refer- 
ence to  the  sun,  is  materially  different  from  that  of  a  superior 
planet.  The  inferior  planets  always  accompany  the  sun,  being 
seen  alternately  on  the  east  and  west  side  of  him,  and  never 
receding  from  him  beyond  a  certain  distance,  while  the  superior 


188  ASTHOICOMT. 

planets  are  seen  at  every  variety  of  angular  distance.  This 
difference  of  apparent  motion  arises  from  tiie  difference  oT  situa- 
tion of  the  orbits  of  an  inferior  and  superior  planet,  with  respect 
to  the  orbit  of  the  earth,  the  one  lying  within  and  tne  other 
without  the  earth's  orbit. 

Let  CAC'B  (Fig.  66)  represent  the  orbit  of  eitlier  one  of  the 
inferior  planets,  Venus  for  example,  and  P  K  T  the  orbit  of  the 
earth  ;  which  we  will  suppose  to  be  circles  and  to  lie  in  the  same 
plane  ;  and  let  M  L  N  represent  the  sphere  of  the  heavens  to  which 
all  bodies  are  referred.  Suppose,  for  the  present,  that  the  earth  is 
stationary  in  the  position  P.  and  through  P  draw  the  lines  P  A, 
P  B,  tangent  to  the  orbit  of  Venus,  and  prolong  them  on  till  they 
intersect  the  heavens  at  a  and  h.  When  Venus  is  at  C,  (the  earth 
being  at  P,)  she  will  be  in  superior  conjunction,  and  when  at  C  in 
inferior  conjunction.  Now,  by  inspecting  the  figure,  it  will  be 
seen  that  in  passing  from  C  to  C,  she  will  be  seen  in  the  heavens 
on  the  east  side  of  the  sun,  and  in  passing  from  C  to  C  on  the 
west  side  of  the  sun  ;  also,  that  in  passing  from  C  to  A  she  will 
recede  from  the  sun  in  the  heavens,  from  A  to  C  approach  him, 
from  C  to  B  recede  from  him  again,  and  from  B  to  C  approach 
him  again,  a  and  h  will  be  her  positions  in  the  heavens  at  the 
times  of  her  greatest  eastern  and  western  elongations. 

When  Venus  is  to  the  east  of  the  sun,  she  is  seen  in  the  even- 
ing, and  called  the  Evening  Star  ;  and  when  to  the  west,  she  is 
seen  in  the  morning,  and  called  the  Morning  Star. 

460.  We  have  in  the  foregoing  investigation  supposed  the 
earth  to  be  stationary,  a  supposition  which  is  contrary  to  the 
fact :  but  it  is  plain  that  the  only  effect  of  the  earth's  motion  in 
the  case  under  consideration,  as  it  is  slower  than  that  of  the 
planet,  is  to  cause  the  points  A,  C,  B  to  advance  in  the  orbit, 
without  altering  the  nature  of  the  apparent  motion  of  the  planet 
with  respect  to  the  sun.  The  orbits  of  the  earth  and  planet  are 
also  ellipses  of  small  eccentricity,  and  are  slightly  inclined  to 
each  other,  instead  of  being  circles  and  lying  in  the  same  plane : 
on  this  account,  as  the  greatest  elongations  will  occur  in  various 
parts  of  the  orbits,  they  will  differ  in  value.  The  greatest  elon- 
gation of  Venus  varies  from  45°  to  47°  12'.  Its  mean  value  is 
about  46°. 

461.  Owing  to  the  circumstance  of  the  orbit  of  Mercury  being 


SYNODIC    REVOLUTION    OP    A    PLANET.  189 

within  the  orbit  of  Venus,  the  greatest  elongation  of  this  planet 
is  less  than  that  of  Venus.  It  varies  between  the  limits  10°  12', 
and  28°  48' ;  and  is,  at  a  mean,  22°  30'. 

462.  Next,  suppose  P  K  T  (Fig.  66)  to  be  the  orbit  of  a  supe- 
rior planet,  and  C  A  C  B  that  of  the  earth  ;  and  as  the  velocity  of 
the  earth  is  much  greater  than  that  of  the  planet,  let  us,  for  the 
present,  regard  the  planet  as  stationary  in  the  position  P,  while 
the  earth  describes  the  circle  C  A  C.  When  the  earth  is  at  C, 
the  planet,  being  at  P,  is  in  conjunction  with  the  sun.  When 
the  earth  is  at  A,  S  A  P  the  elongation  of  the  planet,  is  90°. 
When  it  arrives  at  C  the  planet  is  in  opposition,  or  180°  distant 
from  the  sun.  And  when  it  reaches  B,  the  elongation  is  again 
90°.  At  intermediate  points  the  elongation  will  have  intermedi- 
ate values.  If,  now,  we  restore  to  tlie  planet  its  orbitual  motion, 
we  shall  manifestly  be  conducted  to  the  same  results  relative  to 
the  change  of  elongation,  as  tlie  only  effect  of  such  motion  will 
be  to  throw  the  points  A,  C,  B  forward  in  the  orbit.  It  appears, 
then,  that  in  the  course  of  a  synodic  revolution  a  superior  planet 
will  be  seen  at  all  angular  distances  from  the  sun,  both  on  the 
east  and  west  side  of  him.  From  conjunction  to  opposition,  that 
is,  while  the  earth  is  passing  from  C  to  C,  the  planet  will  be  to 
the  right,  or  to  the  west  of  the  sun  ;  and  will  therefore  be  belov/ 
the  horizon  at  sunset,  and  rise  some  time  in  tlie  course  of  the 
night.  BiU,  from  opposition  to  conjunction,  or  while  the  earth  is 
moving  from  C  to  C,  it  will  be  to  the  east  of  the  sun,  and  there- 
fore above  the  horizon  at  sunset. 

463.  To  find  tlie  length  of  the  synodic  revolution  of  a  flanet. 

Let  us  first  take  an  inferior  planet,  Venus  for  instance.  Sup- 
pose we  assume,  at  a  given  instant,  the  sun,  Venus,  and  the 
earth  to  be  in  the  same  right  line ;  then,  after  any  elapsed  time 
(a  day  for  instance),  Venus  will  have  described  an  angle  w?,  and 
the  earth  an  angle  M  around  the  sun.  Now,  m  is  greater  than 
M ;  therefore  at  the  end  of  a  day,  the  separation  of  Venus  from 
the  earth,  (measuring  the  separation  by  an  angle  formed  by  two 
lines  drawn  from  Venus  and  the  earth  to  the  sun)  will  be 
tn  —  M  ;  at  the  end  of  two  days  (the  mean  daily  motions  contin- 
uing the  same),  the  angle  of  separation  will  be  2  {m  —  M) ;  at  the 
end  of  three  days,  3  {in  —  M) ;  at  the  end  of  s  days,  s  {m  —  M). 
When  the  angle  of  separation  then  amounts  to  360°,  thai  is,  when 


190  ASTRONOMY. 

s  {m  —  M)  =  360°,  the  sun,  Venus,  and  the  earth  must  be  again 
in  the  same  right  hue,  and  in  that  case, 

s  =  _^2i!_  .  .  .  (128). 

In  which  expression  5  denotes  the  mean  duration  of  a  synodic 
revolution,  m  and  M  being  taken  to  denote  the  mean  daily- 
motions. 

We  may  obtain  from  equation  (128)  another  equation,  in  which 
the  synodic  revolution  is  expressed  in  terms  of  the  sidereal  pe- 
riods of  the  earth  and  planet. 

Let  P  and  p  denote  the  sidereal  periods  in  question  ;  then  since 


Id. 

:  M°  : : 

P  :  360°, 

and 

1 

:  m    : : 

p  :  360 ; 

M  = 

360° 
~P^ 

and 
s  = 

360° 

ni  —           ; 

360^ 

substituting 
Fp 

360°  (i  - 
\p 

4) 

V—p 

(129). 


Equations  (128),  (129),  although  investigated  for  an  inferior 
planet,  will  answer  equally  well  for  a  superior  planet ,  provided 
we  regard  ni  as  standing  for  the  mean  daily  motion  of  the  earth, 
M  for  that  of  the  planet,  p  for  the  sidereal  period  of  the  earth, 
and  P  for  that  of  the  planet.  For,  the  earth  holds  towards  a 
superior  planet  the  place  of  an  inferior  planet,  and  a  synodic 
revolution  of  the  earth  to  an  observer  on  the  planet,  will  ob- 
viously be  a  synodic  revolution  of  the  planet  to  an  observer 
on  the  earth. 

464.  Equation  (128)  shows  that  the  length  of  a  mean  synodic 
revolution  depends  altogether  upon  the  amount  of  the  differ- 
ence of  the  mean  daily  motions  of  the  earth  and  planet,  and  is  the 
greater  the  less  is  this  difference. 

It  follows  therefore  that  the  synodic  revolution  is  the  longest 
for  the  planets  nearest  the  earth. 

It  appears  by  equation  (129),  that  the  length  of  a  synodic  revo- 
lution is,  for  an  inferior  planet,  greater  than  the  sidereal  period 
of  the  planet,  and  for  a  superior  planet,  greater  than  the  sidereal 
period  of  the  earth.  The  actual  lengths  of  the  synodic  revolu- 
tions of  the  different  planets  are  given  in  Table  V. 


STATIONS,    &.C.    OF    THE    PLANETS.  191 

465.  The  mean  synodic  revolution  of  a  planet  being  known, 
and  also  the  time  of  one  conjunction  or  opposition,  we  may 
easily  ascertain  its  mean  elongation  at  any  given  time,  and 
thus  approximately  the  time  of  its  rising,  setting,  and  meridian 
passage. 

Stations  and  Retrogradations  of  the  Planets. 

466.  The  apparent  motions  of  the  planets  in  the  heavens,  as 
has  already  been  stated  (Art.  8),  are  not,  like  those  of  the  sun  and 
moon,  continually  from  west  to  east,  or  direct,  but  are  sometimes 
also  from  east  to  west,  or  retrograde.  The  retrograde  motion 
takes  place  over  arcs  of  but  a  small  number  of  degrees  ;  and  in 
changing  the  direction  of  their  motions,  the  planets  are  for  seve- 
ral days  stationary  in  the  heavens.  These  phenomena  are  called 
the  Stations  and  Retrogradations  of  the  planets.  We  now 
propose  to  inquire  theoretically  into  the  particulars  of  the  mo- 
tions in  question,  and  to  show  how  the  phenomena  just  men- 
tioned result  from  the  motions  of  the  planets  in  connection  with 
the  motion  of  the  earth. 

Let  C  A  C  B  (Fig.  66),  represent  the  orbit  of  an  inferior  planet, 
and  P  K  T  the  orbit  of  the  earth  ;  both  considered  as  circles,  and 
as  situated  in  the  same  plane.  If  the  earth  were  continually  sta- 
tionary in  some  point  P  of  its  orbit,  it  is  plain  that  while  the  planet 
was  moving  from  B  the  position  of  greatest  western  elongation, 
to  A  the  position  of  greatest  eastern  elongation,  it  would  advance 
in  the  heavens  from  6  to  a  ;  that,  while  it  was  moving  from  A  to  B, 
that  is,  from  greatest  eastern  to  greatest  western  elongation,  it 
would  retrograde  in  the  heavens  from  aloh]  and  that,  in  passing 
the  points  A  and  B,  as  it  would  be  moving  directly  towards  or 
from  the  earth,  it  would  for  a  time  appear  stationary  in  the  hea- 
vens in  the  positions  a  and  h. 

But  the  earth  is  in  fact  in  motion,  and  the  actual  apparent 
motion  of  the  planet  is  in  consequence  materially  different  from 
this.  Let  A,  A'  (Fig.  67),  be  the  positions  of  the  planet  and  earth 
at  the  time  of  the  greatest  eastern  elono-ation,  C,  P  their  positions 
at  inferior  conjunction,  and  B,  B'  their  positions  at  the  greatest 
western  elongation.  At  the  time  of  the  greatest  eastern  elong-ation 
while  the  planet  describes  a  certain  distance  A  D  on  the  line  of  the 
centres  of  the  earth  and  planet,  the  earth  moves  forward  in  its  or- 


192  ASTRONOMY. 

bit  a  certain  distance  A'  D' ;  so  that,  instead  of  appearing  stationary 
at  a  in  the  interval,  the  planet  will  advance  in  the  heavens  from 
a  to  d.  From  the  same  cause,  it  will  have  a  direct  motion  about 
the  time  of  the  greatest  western  elongation.  As  it  advances  from 
A  towards  C",  the  direct  motion  will  continue ;  but,  as  the  daily- 
arc  described  by  the  planet  will  make  a  less  and  less  angle  with 
the  daily  arc  described  by  the  earth,  the  rate  of  motion  will  con- 
thmally  decrease,  and  finally  when  the  planet  has  come  into  a 
position  with  respect  to  the  earth,  such  that  the  lines  of  direction 
of  the  planet,  m  p,  m'  p',  at  the  beginning  and  end  of  the  day  are 
parallel,  it  will  be  stationary  in  the  heavens.  As  the  daily  arc  of 
the  planet  is  greater  than  that  of  the  earth,  and  becomes  parallel 
to  it  in  inferior  conjunction,  the  planet  will  be  in  the  position  in 
question  before  it  comes  into  inferior  conjunction. 

Subsequent  to  this,  the  inclination  uf  the  daily  arcs  still  dimin- 
ishins:,  the  lines  of  direction  of  the  planet  at  the  beginning  and 
end  of  the  day  will  diverge,  and  therefore  the  motion  will  be  re- 
trograde. After  inferior  conjunction,  the  inclination  of  the  arcs, 
will,  at  corresponding  positions  of  the  earth  and  planet,  obviously, 
be  the  same  as  before.  It  follows  therefore,  that  the  planet  will 
be  at  its  western  station,  when  it  is  at  the  same  angular  distance 
from  the  sun  as  at  its  eastern  station ;  that  its  motion  will  be 
retrograde,  until  it  has  passed  inferior  conjunction  and  arrived  at 
its  western  station  ;  and  that  after  this  it  will  be  direct,  q  and  7i 
represent  the  positions  of  the  planet  and  the  earth  at  the  time  of 
the  western  station  ;  C  q  =  C  p,  and  V  )i  —  P  m. 

The  diminution  of  the  elongation  of  the  planet  at  its  two  sta- 
tions is  not  the  only  effect  of  the  earth's  motion  in  the  case  under 
consideration  ;  it  also  accelerates  the  direct,  and  retards  the  retro- 
grade motion  of  the  planet,  and  gives  to  the  planet  alona;  with 
the  sun  an  apparent  motion  of  revolution  around  the  earth. 

467.  Let  us  now  pass  to  the  case  of  a  superior  planet.  Suppose 
A  C  B  (Fig.  67),  to  be  the  orbit  of  the  earth,  and  A'  P  B'  that  of 
the  planet.  Since  the  earth  is  an  inferior  planet  to  an  observer 
stationed  upon  a  superior  planet,  it  appears  by  the  foregoing  arti- 
cle, that  it  will,  to  an  observer  so  situated,  have  a  retrograde 
motion  while  it  is  passing  over  a  certain  arc  p  C'  q  in  the  infe- 
rior part  of  its  orbit,  and  a  direct  motion  during  the  remainder  of 
the  synodic  revolution.     jNow,  it  is  plain,  that  the  direction  of 


PHASES    OF    THE    INFERIOR    PLANETS.  193 

the  planet's  motion,  as  seen  from  the  earth,  will  always  be  the 
same  as  the  direction  of  the  earth's  motion  as  seen  from  the 
planet.  When  the  earth  is  at  C  the  middle  of  the  arc  p  C  q, 
the  planet  is  in  opposition.  It  follows,  therefore,  that  a  superior 
planet  has  a  retrograde  motion  during  a  small  portion  of  its 
synodic  revolution,  about  the  time  of  opposition. 

Phases  of  the  Inferior'  Planets. 

468.  To  the  naked  sight  the  disc  of  the  planet  Venus  appears 
circular,  like  that  of  each  of  the  other  planets,  but  the  telescope 
shows  this  to  be  an  optical  illusion.  When  Venus  is  repeatedly 
observed  with  a  telescope,  it  is  seen  to  present  in  its  various  posi- 
tions with  respect  to  the  sun  the  same  variety  of  phases  as  the 
moon  ;  being  a  full  circle  at  superior  conjunction,  a  half  circle 
at  the  ofreatest  eastei'ii  and  western  elongations,  and  a  crescent 
with  the  horns  turned  from  the  sun,  before  and  after  inferior 
conjunction. 

469.  Mercury  exhibits  precisely  similar  phases,  but  being 
smaller,  at  a  greater  distance  from  the  earth,  and  much  nearer 
the  sun,  its  phases  are  not  so  easily  observed  as  those  of  Venus. 

470.  The  phases  of  Venus  are  easily  accounted  for,  by  sup- 
posing it  to  be  an  opake  spherical  body,  and  to  shine  by  reflect- 
ing the  sun's  light,  and  by  taking  into  consideration  its  motion 
with  respect  to  the  sun  and  earth.  The  hemisphere  turned 
towards  the  sun  is  illuminated  by  him,  and  the  other  is  in  dark, 
and  as  the  planet  revolves  around  the  sun,  various  portions  of 
the  enlightened  half  are  turned  towards  the  earth ;  in  superior 
conjunction,  the  whole  of  it ;  at  the  greatest  elongations,  one  half; 
and  near  inferior  conjunction,  but  a  small  part.  This  will  be 
abundantly  evident  on  inspecting  Fig.  68.  The  phases  cor- 
responding to  the  positions  represented,  are  delineated  in  the 
figure. 

The  phases  of  Mercury  are  obviously  susceptible  of  a  similar 
explanation. 

471.  The  disc  of  the  planet  Mars  also  undergoes  changes  of 
form,  but  they  are  of  comparatively  moderate  extent.  It  is  some- 
times gibbous,  but  never  has  the  form  of  a  crescent.  Indeed,  on 
the  supposition  that  Mars  is  an  opake  body  illuminated  by  the  sun^ 
we  would  not  see  the  whole  of  the  enlightened  hemisphere,  except 

25 


194  ASTRONOMY. 

in  conjunction  and  opposition,  but  there  would  always  be  more 
than  half  of  it  turned  towards  the  earth,  and  therefore  the  disc 
should  always  be  larger  than  a  half  circle. 

472.  The  discs  of  the  other  superior  planets  do  not  experience 
any  perceptible  variation  of  form,  for  the  reason,  doubtless,  that 
their  orbits  are  so  large  with  respect  to  the  orbit  of  the  earth,  that 
all,  or  very  nearly  all  of  their  illuminated  hemispheres,  is  con- 
stantly visible  from  the  earth. 

Transits  of  the  Inferior  Planets. 

473.  The  two  inferior  planets  Venus  and  Mercury,  at  inferior 
conjunction,  sometimes,  though  rarely,  pass  between  the  sun  and 
earth,  and  are  seen  as  a  dark  spot  crossing  the  sun's  disc.  This 
phenomenon  is  called  a  Transit.  It  will  take  place,  in  the  case 
of  either  planet,  whenever,  at  the  time  of  inferior  conjunction,  it 
is  so  near  either  node  that  its  geocentric  latitude  is  less  than 
the  apparent  semi-diameter  of  the  sun. 

474.  The  transits  of  Venus  take  place  alternately  at  intervals 
of  8  and  105^  or  121^  years.  The  last  were  in  the  years  1761 
and  1769.     The  next  will  be  in  1S74  and  1882. 

475.  In  consequence  of  the  greater  distance  of  Mercury  from 
the  earth,  a  greater  portion  of  its  orbit  is  directly  interposed  be- 
tween the  sun  and  earth,  than  of  the  orbit  of  Venus ;  moreover, 
the  synodic  revolution  of  Mercury  is  shorter  than  that  of  Venus. 
On  these  accounts,  it  happens  that  the  transits  of  Mercury  are 
much  more  frequent  than  those  of  Venus.  The  last  transit  of 
Mercury  was  in  the  year  1835.  The  next  two  will  take  place  in 
1845  and  1848. 

476.  A  transit  is  calculated  in  a  precisely  similar  manner  with 
a  solar  eclipse  ;  the  planet  in  the  one  calculation  answering  to  the 
moon  in  the  other. 

477.  A  transit  is  an  important  phenomenon  in  a  practical  point 
of  view,  as  it  furnishes  the  most  exact  mt  ans  we  possess  of  ascer- 
taining the  sun's  parallax.  In  order  to  understand  how  this  phe- 
nomenon can  be  used  for  this  purpose,  we  have  only  to  consider 
that,  in  consequence  of  the  difference  of  the  parallaxes  of  the  sun 
and  Venus,  observers  at  different  stations  upon  the  earth  will 
refer  the  planet  to  different  points  upon  the  sun's  disc,  and  that 
therefore,  to  such  observers,  the  transit  will  take  place  along  dif- 


/,  ,■ 

APPEARANCES,   DIMENSIONS,  tC.  OF  THE   PLANETS.  195 

ferent  chords,  and  be  accomplished  in  unequal  portions  of  time. 
It  is  then  to  be  expected,  that,  if  the  durations  of  the  transit  at 
two  different  places  should  be  noted,  the  difference  of  the  paral- 
laxes of  the  sun  and  Venus,  upon  which  alone  the  difference  of 
th3  duration  depends,  could  be  computed.  This  com;>utation  is 
in  fact  possible.  Also,  the  ratio  of  the  parallaxes  being  inversely 
as  that  of  the  distances,  could  be  found  by  the  elliptical  theory, 
and  thus  the  parallax  both  of  the  sun  and  Venus  would  become 
known. 

478.  The  parallax  of  the  sun,  as  it  is  now  known,  was  deduced 
from  observations  upon  the  transits  of  Venus  in  1769  and  1761. 
Expeditions  were  fitted  out  on  the  most  efficient  scale,  by  the 
British,  French,  Russian,  and  other  governments,  and  sent  to 
various  parts  of  the  earth,  remote  from  each  other,  to  observe 
the  transit  of  1769,  that  the  parallax  of  the  sun  might  be  com- 
puted from  the  results  of  the  observations.  The  sun's  parallax 
as  determined  by  Professor  Encke  from  the  observations  made 
upon  the  transit  in  question,  and  that  of  1761.  is  8".5776. 

Appearances,  Dimensions,  Rotation,  and  Physical  Constitu- 
tion of  the  Planets. 

479.  It  appears  from  admeasurement  with  the  telescope  and 
micrometer,  that  the  apparent  diameter  of  a  planet  is  subject  to 
sensible  variations.  The  apparent  diameter  of  Venus,  as  well  as 
of  Mercury,  is  greatest  in  inferior  conjunction,  and  least  in  supe- 
rior conjunction  ;  while  the  apparent  diameter  of  each  of  the 
other  planets  is  greatest  in  opposition  and  least  in  conjunction. 
These  variations  of  the  apparent  diameters  of  the  planets,  are 
necessary  consequences  of  the  changes  that  take  place  in  the  dis- 
tances of  the  planets  from  the  earth. 

480.  The  real  diameter  of  a  planet  is  deduced  from  its  appa- 
rent diameter  and  horizontal  parallax.  (See  Art.  395.)  When 
the  diameters  of  the  planets  have  been  found,  their  relative  sur- 
faces and  volumes  are  easily  obtained ;  for  the  surfaces  are  as  the 
squares  of  the  diameters,  and  the  volumes  as  the  cubes. 

481.  The  order  of  magnitude  of  the  planets  is  as  follows : 
1  Jupiter,  2  Saturn,  3  Uranus,  4  the  Earth,  5  Venus,  6  Mars, 
7  Mercury,  8  Pallas,  9  Ceres,  10  Juno,  11  Vesta.  The  range  of 
magnitude,  for  the  principal  planets,  is  from  1  to  21000. 


196  ASTRONOMY. 

482.  Spots  more  or  less  dark  have  been  seen  upon  the  discs  of 
most  of  the  principal  planets  ;  and  by  passino-  across  them  from 
east  to  west  and  re-appearing  at  the  eastern  limbs,  have  estab- 
lished that  the  planets  upon  which  they  are  observed,  rotate  upon 
axes  from  west  to  east.  From  repeated  careful  observations 
upon  the  situations  of  these  spots,  the  periods  of  rotation,  and 
the  positions  of  the  axes,  have  been  determined. 

The  periods  of  rotation  of  Mercury,  Venus,  the  Earth,  and 
Mars,  are  all  about  24  hours,  and  of  Jupiter  and  Saturn  about  10 
hours.  Those  of  the  other  planets  are  not  known.  The  axes 
of  rotation  remain  continually  parallel  to  themselves,  as  the 
planets  revolve  in  their  orbits. 

483.  The  amount  of  light  and  heat,  which  the  sun  bestows  upon 
the  planets,  decreases  as  we  recede  from  the  sun,  in  the  same  ratio 
that  the  square  of  the  distance  increases.     (See  Table  IV.) 

Mercury. 

484.  In  consequence  of  its  proximity  to  the  sun,  Mercury  is 
rarely  visible  to  the  naked  eye.  When  seen,  it  presents  the 
appearance  of  a  star  of  the  3d  or  4th  magnitude.  Its  phases 
show  that  it  is  opake,  and  illuminated  by  the  sun.  Its  apparent 
diameter  varies  with  its  distance  from  5"  to  12".  Its  real  diam- 
eter is  about  3140  miles,  or  |  of  that  of  the  earth,  and  its  vol- 
ume is  about  y'g  of  the  earth's  volume.* 

Mercury  performs  a  rotation  on  its  axis  in  24  h.  5^m.,  and  its 
axis  is  inclined  to  the  ecliptic  under  a  small  angle.  It  is  believed 
to  be  surrounded  by  a  dense  atmosphere. 

Venus. 

485.  Venus  is  the  most  brilliant  of  all  the  planets,  and  gene- 
rally appears  larger  and  brighter  than  any  of  the  fixed  stars.  At 
times,  it  emits  so  much  light  as  to  be  visible  in  the  open  day,  if 
the  eye  be  protected  from  the  sun.  It  is  found  by  calculation, 
that  the  epochs  in  the  course  of  a  synodic  revolution,  at  which 
Venus  gives  most  light  to  the  earth,  are  those  at  which,  being  in 
the  inferior  part  of  its  orbit,  it  has  an  elongation  of  about  40°. 
They  are  about  36  days  before  and  after  inferior  conjunction. 
The  disc  is  then  considerably  less  than  a  semi-circle,  but  the 


*  The  exact  diameters,  volumes,  times  of  rotation,  &.C.,  of  the  different  planets, 
as  far  as  known,  may  bo  found  in  Table  IV. 


VENUS — MARS.  197 

increased  proximity  to  the  earth  more  than  compensates  for  the 
diminished  size  of  the  disc.  Venus  will  besides  attain  to  greater 
splendour  in  some  revolutions  than  others,  in  consequence  of 
being  nearer  the  earth,  when  in  the  most  favorable  position. 

486.  As  seen  through  a  telescope,  Venus  presents  a  disc  of 
nearly  uniform  brightness,  and  spots  have  very  rarely  been 
seen  upon  it.  Its  phases  prove  it  to  be  an  opake  spherical  body, 
shining  by  reflecting  the  sun's  light.  Its  apparent  diameter 
varies  with  its  distance  from  10"  to  61".  Its  real  diameter  is 
about  7700  miles,  and  its  volume  about  -j-^^  less  than  that  of  the 
earth.  The  period  of  its  rotation  is  23h.  21m.  The  inclination 
of  its  axis  to  the  plane  of  its  orbit  is  not  exactly  known,  but  is 
not  far  from  18".  Schroeter  inferred,  from  a  gradual  diminu- 
tion of  its  light  observed  at  the  edge  of  its  disc,  that  it  was  sur- 
rounded by  an  atmosphere  analogous  to  our  own. 

Mars. 

487.  Mars  is  of  the  apparent  size  of  a  star  of  the  first  or  second 
magnitude,  and  is  distinguished  from  the  other  planets  by  its 
red  and  fiery  appearance.  The  observed  variation  in  the  form 
of  its  disc  (Art.  471),  shows  that  it  derives  its  light  from  the  sun. 
Its  greatest  and  least  apparent  diameters  are  respectively  4"  and 
18".  Its  real  diameter  is  about  4100  miles,  or  rather  more  than 
^  of  the  diameter  of  the  earth,  and  its  bulk  is  about  ^  of  that  of 
the  earth. 

Mars  revolves  on  its  axis  in  24h.  39m.  ;  and  its  axis  is  in- 
clined to  the  ecliptic  in  an  angle  of  about  GO'^.  It  appears,  from 
measurements  made  with  the  micrometer,  that  its  polar  diameter 
is  less  than  the  equatorial,  and  thus,  that,  like  the  earth,  it  is 
flattened  at  its  poles.  According  to  Sir  W.  Herschel,  its  oblate- 
ness  (Art.  145)  is  j\. 

488.  Spots  of  dilferent  shades  are  generally  visible  upon  the 
surface  of  Mars,  most  of  which  always  present  the  same  appear- 
ance whenever  they  are  distinctly  seen. 

The  ruddy  colour  of  the  light  of  Mars  has  been  generally  at- 
tributed to  a  dense  atmosphere  surrounding  the  planet.  Sir  J. 
F.  W.  Herschel,  however,  has  observed,  in  examining  this  planet 
with  a  good  telescope,  that  some  of  its  spots  are  of  a  reddish 
colour,  and  thence  concludes  that  the  ruddy  colour  of  its  light 
is  owing  to  a  red  tinsfe  in  its  soil. 


198  ASTRONOMY. 

Jupiter  and  its  Satellites. 

489.  Jupiter  is  the  most  brilliant  of  the  planets,  except  Venus, 
and  sometimes  even  surpasses  Venus  in  brightness.  The 
eclipses  of  its  satellites  prove  that  it  is  an  opake  body,  and  that  it 
shines  by  reflectinof  the  light  of  the  sun.  Its  apparent  diameter 
when  greatest,  is  46",  and  when  least,  30". 

Jupiter  is  the  largest  of  all  the  planets.  Its  diameter  is  nearly 
11  times  the  diameter  of  the  earth,  or  about  86,000  miles,  and 
its  bulk  is  nearly  1300  times  that  of  tho  earth.  It  turns  on  an 
axis  nearly  perpendicular  to  the  ecliptic,  and  completes  a  rota- 
tion in  9h.  56m.  The  polar  diameter  is  about  -^\  less  than  the 
equatorial. 

490.  When  Jupiter  is  examined  with  a  good  telescope,  its 
disc  is  always  observed  to  be  crossed  by  several  obscure  spaces, 
which  are  nearly  parallel  to  each  other,  and  to  the  plane  of  the 
equator.  These  are  called  the  Belts  of  Jupiter.  They  are 
generally  confined  to  the  immediate  vicinity  of  the  equator,  but 
they  sometimes  also  extend  to  considerable  distances  from  it, 
and  have  even  been  seen  distributed  over  the  whole  face  of  the 
planet. 

491.  The  satellites  of  Jupiter,  as  it  has  been  already  re- 
marked, are  visible  with  telescopes  of  moderate  power.  With 
the  exception  of  the  second,  which  is  a  little  smaller,  they  are 
somewhat  larger  than  the  moon.  The  orbits  of  the  satellites 
lie  very  nearly  in  the  plane  of  Jupiter's  equator. 

492.  Sir  W.  Herschel,  in  examining  the  satellites  of  Jupiter 
with  a  telescope,  perceived  that  they  underwent  periodical  vari- 
ations of  brightness.  These  variations  he  supposed  to  proceed 
from  a  rotation  of  the  satellites  upon  axes,  Avhich  caused  them 
to  turn  different  faces  towards  the  earth  ;  and  from  repeated  and 
careful  observations  made  upon  them,  he  discovered  that  each 
satellite  made  one  turn  upon  its  axis  in  the  same  time  that  it 
accomplished  a  revolution  around  the  primary  ;  and,  therefore, 
like  the  moon,  presented  continually  the  same  face  to  the 
primary. 

Saturn,  with  its  Satellites  and  Ring: 

493.  Saturn  shines  with  a  pale  dull  light.  Its  apparent  dia- 
meter varies  only  3"  or  4"  by  reason  of  the  change  of  distance, 


SATURN.  199 

and  is  at  the  mean  distance  about   16".     The  edipses  of  its 
satellites  prove  that  it  is  opake,  and  illuminated  by  the  sun. 

Saturn  is  the  largest  of  the  planets,  next  to  Jupiter.  Its  dia- 
meter is  nearly  10  times  the  diameter  of  the  earth,  or  79.000 
miles ;  and  its  volume  is  nearly  1000  times  that  of  the  earth. 
The  rotation  on  its  axis  is  performed  in  lOh.  29m.  The  incli- 
nation of  its  axis  to  the  ecliptic  is  about  60°.  Its  oblateness 
is  tV- 

494.  The  disc  of  Saturn,  like  that  of  Jupiter,  is  frequently 
crossed  with  dark  bands  or  belts,  in  a  direction  parallel  to  its 
equator.  Extensive  dusky  spots  are  also  occasionally  seen 
upon  its  surface. 

495.  The  planet  Saturn  is  distinguished  from  all  the  other 
planets  in  being  surrounded  by  a  broad,  thin,  luminous  ring,  situ- 
ated in  the  plane  of  its  equator,  and  entirely  detached  from  the 
body  of  the  planet.  This  ring  sometimes  casts  a  shadow  upon 
the  planet,  and  is  in  turn,  at  times,  partially  obscured  by  the 
shadow  of  the  planet ;  from  which  we  conclude  that  it  is  opake, 
and  receives  its  light  from  the  sun. 

It  is  inclined  to  the  plane  of  the  ecliptic  in  an  angle  of  about 
30°,  and  during  the  motion  of  Saturn  in  its  orbit,  it  remains  con- 
tinually parallel  to  itself  The  face  of  the  ring  is,  therefore, 
never  viewed  perpendicularly  from  the  earth,  and  for  this  reason 
never  appears  circular,  although  such  is  its  actual  form.  Its  ap- 
parent form  is  that  of  an  ellipse,  more  or  less  eccentric,  according 
to  the  obliquity  under  which  it  is  viewed,  which  varies  with  the 
position  of  Saturn  in  its  orbit.  When  it  is  seen  under  the  larger 
cUigles  of  obliquity,  it  appears  as  a  luminous  band,  nearly  encir- 
cling the  planet,  and  is  visible  in  telescopes  of  small  power.  Stars 
can  also  be  seen  between  it  and  the  planet  in  these  positions.  At 
other  times,  when  viewed  very  obliquely,  it  can  be  seen  only  with 
telescopes  of  high  power.  When  it  is  approaching  the  latter  state, 
it  has  the  appearance  of  two  handles  or  atiscB,  one  on  each  side  of 
the  planet. 

It  is  also  at  times  invisible.  This  is  the  case  whenever  the 
earth  and  sun  are  on  different  sides  of  the  plane  of  the  ring,  for 
the  reason  that  the  illuminated  face  is  then  turned  from  the  earth. 
When  the  plane  of  the  ring  passes  through  the  centre  of  the  sun, 
the  illuminated  edge  can  be  seen  only  in  telescopes  of  extraordi- 


200  ASTHUNOMY. 

nary  power,  and  appears  as  a  thread  of  light  cutting  the  disc  of  the 
planet. 

496.  Since  the  orbit  of  Saturn  is  very  large  in  comparison 
with  the  orbit  of  the  earth,  the  plane  of  the  ring  will,  during  the 
greater  part  of  the  revolution  of  Saturn,  pass  without  the  orbit  of 
the  earth ;  and  when  this  is  the  case,  the  ring  will  be  visible,  as 
the  earth  and  sun  will  be  on  the  same  side  of  its  plane.  During 
the  period,  which  is  about  a  year,  that  the  plane  of  the  ring  is 
passing  by  the  orbit  of  the  earth,  the  earth  will  sometimes  be  on 
the  same  side  of  it  as  the  sun,  and  sometimes  on  opposite  sides.  In 
the  former  case  the  ring  will  be  invisible,  and  in  the  latter  will  be 
seen  so  obliquely  as  to  be  visible  only  in  telescopes  of  considerable 
or  great  power.  All  this  will  perhaps  be  better  understood  on 
consulting  Fig.  69,  where  ABC  represents  the  orbit  of  Saturn, 
E  F  G  that  of  the  earth,  and  P,  d,  N,  R,  S,  diflferent  positions  of  the 
ring. 

The  plane  of  the  ring  will  pass  through  the  sun  every  semi- 
revolution  of  Saturn,  or,  at  a  mean,  about  every  15  years,  and 
at  the  epochs  at  which  the  longitude  of  the  planet  is  respectively 
170^  and  350°,  these  being  the  longitudes  of  the  nodes  of  the  ring. 
The  ring  will  then  disappear  once  in  about  15  years ;  but,  owing 
to  the  different  situations  of  the  earth  in  its  orbit,  under  circum- 
stances oftentimes  quite  different.  And  the  disappearance  will 
occur  when  the  longitude  of  the  planet  is  about  170°,  or  350°, 
The  ring  will  be  seen  to  the  greatest  advantage  when  the  longi- 
tude of  the  planet  is  not  far  from  80°  or  260°.  The  last  disap- 
pearance took  place  in  1833  ;  the  next  will  be  in  1848.  At  the 
present  time  (1838)  the  north  face  of  the  ring  is  visible. 

497.  From  observations  made  upon  bright  spots  seen  on  the  face 
of  the  ring,  Herschel  discovered  that  it  revolved  from  west  to  east 
about  an  axis  perpendicular  to  its  plane,  and  passing  through  the 
centre  of  the  planet,  (or  very  nearly).  The  period  of  its  rotation  is 
lOh.  29m.  It  is  remarkable  that  this  is  the  period  in  which  a  satel- 
lite assumed  to  be  at  a  mean  distance  equal  to  the  mean  distance 
of  the  particles  of  the  ring,  would  revolve  around  the  primary 
according  to  the  third  law  of  Kepler. 

The  breadth  of  the  ring  is  about  one  half  greater  than  its  dis- 
tance from  the  surface  of  the  planet,  and  is  about  equal  to  one 
third  the  diameter  of  the  planet,  or  29,000  miles. 


SATURN L'KAKUaJ VKSTA — JUNO — CERES — PALLAS.       201 

498.  What  we  have  called  Saturn's  ring,  consists  in  fact  of  two 
concentric  rings,  which  turn  together,  although  entirely  detached 
from  each  other.  The  void  space  between  them  is  perceived  in 
telescopes  of  high  power,  under  the  form  of  a  black  circular  line. 
According  to  the  calculations  of  Sir  John  Herschel,  from  the  micro- 
metric  measures  of  Prof  Struve,  the  breadth  of  the  interior  ring  is 
about  17200  miles,  and  of  the  exterior  about  10600  miles  ;  the  in- 
terval between  the  rings  is  about  1800  miles,  and  the  distance 
from  the  planet  to  the  inside  of  the  interior  ring  is  about  19000 
miles.  The  thickness  of  the  rings  is  not  well  Imown,  the  edge 
subtends  an  angle  less  than  1",  which,  at  the  distance  of  the  planet, 
answers  to  about  4000  miles.  Herschel  makes  it  less  than  100 
miles. 

499.  The  satellites  of  Saturn,  with  the  exception  of  the  7th, 
revolve  very  nearly  in  the  plane  of  the  ring.  The  orbit  of  the 
7th  is  inclined  about  30°  to  this  plane. 

The  7th  satellite  is  by  far  the  largest  and  most  conspicuous. 
The  1st  and  2d,  which  just  skirt  the  edge  of  the  ring,  have  only 
been  seen  by  Sir  William  Herschel  in  his  large  telescope. 

The  7th  satellite  is  subject  to  periodical  variations  of  lustre, 
which  prove  its  rotation  on  an  axis  in  the  period  of  a  sidereal 
revolution  of  Saturn. 

Uranus  and  its  Satellites. 

500.  Uranus  is  scarcely  visible  to  the  naked  eye.  In  a  tele- 
scope it  appears  as  a  small  round  uniformly  illuminated  disc.  Its 
apparent  diameter  is  about  4",  from  which  it  never  varies  much, 
owing  to  the  smallness  of  the  earth's  orbit  in  comparison  with  its 
own.  Its  real  diameter  is  about  34000  miles,  and  its  bulk  80  times 
that  of  the  earth.  Analogy  leads  us  to  believe  that  this  planet  is 
opake  and  turns  on  an  axis,  but  there  is  no  direct  proof  that  this 
is  the  case. 

501.  The  satellites  of  Uranus  were  discovered  by  Sir  W.  Her- 
schel. They  are  discernible  only  with  telescopes  of  the  highest 
power. 

Vesta — Juno —  Ceres — Pallas. 

502.  These  four  planets,  although  less  distant  than  several  of 
the  others,  are  so  extremely  small,  that  they  can  only  be  seen 
with  telescopes  of  considerable  power. 

503.  The  magnitudes  of  these  planets  are  not  well  known. 

26 


202  ASTKUNOMV. 

Vesta  is  the  smallest  and  also  the  most  brilliant.  Ceres  and  Pal- 
las are  said  to  have  a  nebulous  or  hazy  appearance,  indicative  of  an 
extensive  vaporous  atmosphere. 


CHAPTER    XVIII. 

OF    COMETS. THEIR    APPEARANCE,    MAGNITUDE,    AND 

PHYSICAL    CONSTITUTION. 

504.  A  comet  usually  consists  of  a  mass  of  some  luminous 
vapoury  substance,  called  the  ComUj  condensed  towards  its  centre 
around  a  brilliant  Nucleus  that  is  in  general  not  very  distinctly 
defined,  from  which  diverges  in  a  direction  opposite  to  the  sun,  a 
stream  of  luminous  transparent  vapour,  called  the  Tail.  The 
coma  and  nucleus  together  form  what  is  called  the  Head  of  the 
comet, 

505.  The  length  and  form  of  the  tail  are  very  various.  In 
some  instances  it  is  only  a  few  degrees  in  length,  and  in  others  it 
is  more  than  a  quadrant.  The  tail  of  the  great  comet,  which 
appeared  in  1680,  extended  to  a  distance  of  .70°  from  the  head, 
and  that  of  the  comet  of  1618,  to  104°. 

506.  When  a  comet  first  appears,  no  tail  is  perceptible,  and  its 
light  is  very  faint.  As  it  approaches  the  sun,  it  becomes  brighter ; 
the  tail  also,  after  a  time,  shoots  out  from  the  coma,  and  increases 
from  day  to  day  in  extent  and  distinctness.  As  the  comet 
recedes  from  the  sun,  the  tail  precedes  the  head,  being  still  on  the 
opposite  side  from  the  sun.  and  grows  less  and  less,  at  the  same 
time  that,  along  with  the  head,  it  decreases  in  brightness,  till  at 
length  the  comet  resumes  nearly  its  first  appearance. 

The  tail  of  a  comet  is  the  longest,  and  the  whole  comet  is  in- 
trinsically the  most  luminous,  soon  after  it  has  passed  its  perihe- 
lion.    Its  apparent  size  and  lustre  will  not,  however,  generally  be 


APPEARANCE,  MAGNITUnE,   AND   DENSrTY   OF  COMETS.       203 

the  greatest  at  this  time,  as  tliey  will  depend  upon  the  distance 
and  position  of  the  earth,  as  well  as  the  actual  size  and  intrinsic 
brightness  of  the  comet. 

507.  Individual  comets  offer  considerable  varieties  of  aspect. 
Some  comets  have  been  seen  which  were  wholly  destitute  of  a 
tail :  such,  among  others,  was  the  comet  of  1682.  Others  have 
had  more  than  one  of  these  appendages.  The  comet  of  1744 
had  six,  which  were  spread  over  an  extent  of  117°;  and  that  of 
1824  two,  the  one  directed  towards  the  sun,  the  other  from  him. 
Others  still  are  without  a  nucleus,  as  the  comet  of  1795. 

The  comets  that  are  visible  only  in  telescopes,  which  are  very 
numerous,  have  generally  no  distinct  nucleus,  and  are  unprovided 
with  a  tail.  They  have  the  appearance  of  round  masses  of  lumi- 
nous vapour,  somewhat  more  dense  towards  the  centre.  Such 
are  Encke's  and  Biela's  comets. 

508.  Comets  are  the  most  voluminous  bodies  in  the  solar 
system.  The  tail  of  the  great  comet  of  1G80,  was  found  by 
Newton  to  have  been,  when  longest,  no  less  than  123,000,000 
miles  in  length.  Some  other  comets  have  had  tails  of  nearly  as 
great  length.  The  heads  of  comets  are  usually  many  thousand 
miles  in  diameter.  That  of  the  comet  of  1811  had  a  diameter  of 
540000  miles. 

509.  The  quantity  of  matter  which  enters  into  the  constitution 
of  a  comet  is  exceedingly  small.  This  is  proved  by  the  fact  that 
the  comets  have  had  no  influence  upon  the  motions  of  the  planets 
or  satellites,  although  they  have  in  many  instances  passed  near 
these  bodies.  The  comet  of  1770,  which  was  quite  large  and 
bright,  passed  through  the  midst  of  Jupiter's  satellites  without 
deranging  their  motions  in  the  least  perceptible  degree.  It  also 
appears  that  the  cometic  matter  is  very  rare  and  subtle,  from  the 
circumstance  of  stars  of  small  magnitude  being  visible  through 
all  parts  of  the  comet,  with  perhaps  the  single  exception  of  the 
nucleus. 

510.  Of  the  physical  constitution  of  the  comets,  little  is  known. 
It  is  not  yet  fully  ascertained  whether  the  vapoury  substance  of 
the  coma  and  tail  is  self-luminous,  or  is  illuminated  by  the  sun. 
The  nucleus  is  by  some  astronomers  supposed  to  be,  in  some  in- 
stances, a  solid,  and  by  others  to  be,  in  all  cases,  a  highly  con- 
densed vapour.     It  has,  in  the  case  of  a  few  cornels,  presented  a 


204  ASTRONOMY. 


well-defined  disc,  like  a  solid,  but  it  has,  in  no  one  instance,  ex- 
hibited phases,  which  it  is  to  be  expected  it  would,  if  it  were  a 
solid  shining  by  reflecting  the  light  of  the  sun. 


CHAPTER    XIX. 

OF    THE    FIXED    STARS. THEIR    NUMBER,    AND    DISTRIBUTION 

OVER     THE    HEAVENS — ANNUAL    PARALLAX,    AND    DISTANCE 

VARIABLE    STARS DOUBLE    STARS — CLUSTERS    OF    STARS, 

AND    NEBULAE. 

511.  The  number  of  stars  that  can  be  seen  with  the  naked  eye, 
does  not  much  exceed  3000,  and  it  is  generally  stated  that  not 
many  more  than  1000  are  ever  visible  at  any  one  time  to  the 
naked  eye.  But  the  telescope  brings  into  view  many  mil- 
lions, and  every  improvement  made  in  it  greatly  increases  the 
number. 

512.  As  to  the  number  of  stars  belonging  to  each  diflferent 
magnitude,  astronomers  assign  from  15  to  20  to  the  first  magnitude, 
from  30  to  60  to  the  second,  about  200  to  the  third,  and  so  on  ; 
the  numbers  increasing  very  rapidly  as  we  descend  in  the  scale  of 
brightness ;  the  whole  number  of  stars  already  registered  down 
to  the  seventh  magnitude,  inclusive,  amounting  to  15000  or 
20000. 

513.  It  is  not  to  be  understood  that  the  classification  of  the  stars 
into  different  magnitudes  is  made  according  to  any  fixed  definite 
proportion  subsisting  between  the  degrees  of  apparent  brightness 
of  the  stars  belonging  to  diflferent  classes.  Stars  of  almost  every 
gradation  of  brightness,  between  the  highest  and  the  lowest,  are 
met  with.  Those  which  offer  marked  differences  of  lustre,  form 
the  basis  of  the  classification  ;  others,  which  do  not  differ  very 
widely  from  these,  are  united  to  them.  As  a  necessary  conse- 
quence, there  are  some  stars  of  intermediate  lustre,  which  cannot 


ANNUAL    PARALLAX    OF    THE    FIXED    STARS.  205 

be  assioned  with  certainty  to  either  magnitude.  Thus,  in  the 
catalogue  of  the  Astronomical  Society  of  London,  3  stars  are 
marked  as  intermediate  between  the  first  and  second  magnitudes, 
and  29  between  the  second  and  third. 

514.  As  to  the  proportions  of  light  emitted  from  the  average 
stars  of  the  different  magnitudes,  according  to  the  experimental 
comparisons  of  Sir  Wm.  Herschel,  they  are,  from  the  first  to  the 
sixth  magnitude,  in  the  ratio  of  the  numbers,  100,  25,  12,  6,  2,  1. 

515.  With  the  exception  of  the  three  or  four  brightest  classes, 
the  stars  are  not  distributed  indiscriminately  over  the  sphere  of 
the  heavens,  but  are  accumulated  in  far  greater  numbers  on  the 
borders  of  the  milky  way,  and  in  the  milky  way  itself,  which  the 
telescope  shows  to  consist  of  an  immense  number  of  stars  of  small 
magnitude  in  close  proximity. 

Annual  Parallax  and  Distance  of  the  Stars. 

516.  The  Annual  Parallax  of  a  fixed  star  is  the  angle  made 
by  two  lines  conceived  to  be  drawn,  the  one  from  the  sun  and  the 
other  from  the  earth,  and  meeting  at  the  star,  at  the  time  the  earth 
is  in  such  part  of  its  orbit  that  its  radius  vector  is  perpendicular  to 
the  latter  line  ;  or,  in  other  words,  it  is  the  greatest  angle  that  can 
be  subtended  at  the  star  by  the  radius  of  the  earth's  orbit.  Thus, 
let  S  (Fig.  70)  be  the  sun,  5  a  fixed  star,  and  E  the  earth, 
in  such  a  position  that  the  radius  vector  S  E  is  perpendicular  to 
the  line  of  direction  of  the  star,  E  s  ;  then  the  angle  S  5  E  is  the 
annual  parallax  of  the  star  5. 

517.  If  the  annual  parallax  of  a  star  was  known,  we  might 
easily  find  its  distance  from  the  earth  ;  for,  in  the  right  angled 
triansfle  S  E  5  we  would  know  the  angle  S  s  E  and  the  side  S 
E,  and  we  should  only  have  to  compute  the  side  E  s.  Now,  if 
any  of  the  fixed  stars  have  a  sensible  parallax,  it  could  be  detected 
by  a  comparison  of  the  places  of  the  star,  as  observed  from  two 
positions  of  the  earth  in  its  orbit,  diametrically  to  each  other ; 
and  accordingly,  the  attention  of  astronomers  furnished  with  the 
most  perfect  instruments,  has  long  been  directed  to  such  observa- 
tions upon  the  places  of  some  of  the  fixed  stars  which  were  sup- 
posed to  be  the  nearest,  in  order  to  determine  their  annual  paral- 
lax. But,  after  exhausting  every  refinement  of  observation,  they 
have  not  been  able  to  establish  that  any  of  them  have  a  measura- 


206  ASTKONOMY. 

ble  parallax.  Now,  such  is  the  nicety  to  which  the  observations 
have  been  carried,  that,  did  the  angle  in  question  amount  to  as 
much  as  1",  it  could  not  possibly  have  esca])ed  detection  and 
universal  recognition.  We  may  then  conclude,  that  the  annual 
parallax  of  the  nearest  fixed  star  is  less  than  1". 

518.  Taking  the  parallax  at  1",  the  distance  of  the  star  comes 
out  206265  times  the  distance  of  the  sun  from  the  earth, 
or  about  20  billions  of  miles.  The  distance  of  the  nearest  fixed 
star  must  therefore  be  greater  than  this.  A  juster  notion  of  the 
immense  distance  of  the  fixed  stars,  than  can  be  conveyed  by 
figures,  may  be  gained  irom  the  consideration  that  light,  which 
traverses  the  distance  between  the  sun  and  earth  in  8m.  13s., 
and  would  perform  the  circuit  of  our  globe  in  |  of  a  second,  em- 
ploys more  than  three  years  in  coming  from  the  nearest  fixed 
star  to  the  earth. 

519.  The  amount  of  light  received  from  the  same  body  at  dif- 
ferent distances,  varies  inversely  as  the  square  of  the  distance. 
Hence,  if  we  admit  the  lisfht  of  a  star  of  each  magnitude  to  be 
half  that  of  one  of  the  next  higher  magnitude,  a  star  of  the  first 
magnitude  would  have  to  be  removed  to  181  times  its  distance, 
to  appear  no  brighter  than  one  of  the  sixteenth.  Accordingly, 
if  the  difference  in  the  apparent  magnitude  of  the  stars  arises  for 
the  most  part  from  a  difference  of  distance,  (which  is  the  more 
probable  supposition),  there  must  be  a  multitude  of  stars  visible 
in  telescopes,  the  light  of  which  has  taken  at  least  five  hundred 
years  to  reach  the  earth. 

Variable  Stars. 

520.  Several  of  the  fixed  stars  are  subject  to  periodical 
changes  of  brightness,  and  are  hence  called  Variable  iStars,  or 
Periodical  Stars.  One  of  the  most  remarkable  of  the  variable 
stars  is  the  star  Omicroft,  in  the  constellation  Cetus.  From 
beinof  as  brigfht  as  a  star  of  the  second  magnitude,  it 
gradually  decreases,  until  it  entirely  disappears :  and,  after  re- 
maining for  a  time  invisible,  re-appears,  and  gradually  increas- 
ing in  lustre,  finally  recovers  its  original  appearance.  The 
period  of  these  changes  is  334  days.  It  remains  at  its  greatest 
brightness  about  two  weeks,  employs  about  three  months  in 
waning  to  its  disappearance,  continues  invisible  for  about  five 


VARIABLE  OK    PERIODICAL  STARS.  207 

months,  and  during  the  remaining-  three  months  of  its  period 
increases  to  its  original  lustre.  Such  is  the  g-eneral  course  of 
its  phases.  It  does  not,  however,  always  recover  the  same  de- 
gree of  brightness,  nor  increase  and  diminish  by  the  same  gra- 
dations. And  it  is  related  by  Hevelius,  that  in  one  instance  it 
remained  invisible  for  a  period  of  four  years,  viz  :  from  October, 
1672,  to  December,  1676. 

521.  The  greater  number  of  variable  stars  undergo  a  regular 
increase  and  diminution  of  lustre,  without  ever,  like  the  star  just 
noticed,  becoming  entirely  invisible.  The  star  Algol,  or  /3  Per- 
seii,  is  a  remarkable  variable  star  of  this  description.  For  a 
period  of  2d.  14h.  it  fippears  as  a  star  of  the  second  magnitude, 
after  which  it  suddenly  begins  to  diminish  in  splendour,  and  in 
about  3^  hours  is  reduced  to  a  star  of  the  fourth  magnitude.  It 
then  begins  again  to  increase,  and  in  3^  hours  more  is  restored 
to  its  usual  brightness,  going  through  all  its  changes  in  2d. 
20h.  48m. 

522.  There  are  also  some  instances  on  record  of  temporary 
stars  having  made  their  appearance  in  the  heavens ;  breaking 
forth  suddenly  in  great  splendour,  and  without  changing  their 
positions  among  the  other  stars,  after  a  time  entirely  disappear- 
ing. One  of  the  most  noted  of  these  is  the  star  which  suddenly 
shone  forth  with  great  brilliancy  on  the  11th  of  November, 
1572,  in  the  constellation  Cassiopeia,  and  was  attentively  ob- 
served by  Tyclio  Brahe,  a  celebrated  Danish  astronomer.  It 
was  then  as  bright  as  any  of  the  permanent  stars,  and  continued 
to  increase  in  splendour  till  it  surpassed  Jupiter  when  brig-htest, 
and  was  visible  at  mid-day.  It  began  to  diminish  in  December 
of  the  same  year,  and  in  March  1574  it  entirely  disappeared, 
after  having  remained  visible  for  sixteen  months,  and  has  not 
since  been  seen. 

In  the  years  945  and  1264,  brilliant  stars  appeared  in  the 
same  regions  of  the  heavens.  It  is  conjectured  from  the  tolera- 
bly near  agreement  of  the  intervals  of  the  appearance  of  these 
stars  and  that  of  1572,  that  the  three  may  be  one  and  the  same 
star,  with  a  period  of  about  300  years.  The  places  of  the  stars 
of  945  and  1264  are,  however,  too  imperfectly  known  to 'estab- 
lish this  with  any  degree  of  certainty. 

523.  What  is  no  less  remarkable  than  the  changes  we  have 


208  ASTRONOMY. 

noticed,  several  stars,  which  are  mentioned  by  tlie  ancient  astro- 
nomers, have  now  ceased  to  be  visible,  and  some  are  now  visible 
to  the  naked  eye  which  are  not  in  the  ancient  catalogues. 

Double  Stars. 

524.  Many  of  the  stars  which  to  the  naked  eye  appear  single, 
when  examined  with  telescopes  are  found  to  consist  of  two  (in 
some  instances  three)  stars  in  close  proximity  to  each  other. 
These  are  called  Double  Stars.  This  class  of  bodies  were  first 
attentively  observed  by  Sir  William  Herschel,  who,  in  the  years 
1782  and  1785,  published  catalogues  of  a  large  number  of 
them  which  he  had  observed.  The  list  has  since  been  greatly 
increased  by  Professor  Struve,  of  Dorpat,  Sir  J.  F.  W.  Herschel, 
and  other  observers. 

525.  Double  stars  are  of  various  degrees  of  proximity.  In 
a  great  number  of  instances,  the  angular  distance  of  the 
individual  stars  is  less  than  1",  and  the  two  can  only  be  sepa- 
rated by  the  most  powerful  telescopes.  In  other  instances, 
the  distance  is  from  1'  to  2',  and  the  separation  can  be  effected 
with  telescopes  of  very  moderate  power.  They  are  divided 
into  six  different  classes,  according  to  their  distances,  those  in 
which  the  proximity  is  the  closest  forming  the  first  class. 

526.  Sir  William  Herschel  instituted  a  series  of  observations 
upon  several  of  the  double  stars,  with  the  view  of  ascertaining 
whether  the  apparent  relative  situation  of  the  individual  stars 
experienced  any  change,  in  consequence  of  the  annual  varia- 
tion of  the  parallax  of  the  star.  With  micrometei-s  adapted  to 
the  purpose,  he  measured  from  time  to  time  the  apparent  dis- 
tance of  the  two  stars,  and  the  angle  formed  by  their  line  of 
junction  with  the  meridian  at  the  time  of  the  meiidian  passage, 
called  the  Angle  of  Position.  Instead,  however,  of  finding  that 
annual  variation  of  these  angles,  which  the  parallax  of  the 
earth's  annual  motion  would  produce,  he  observed  that,  in 
many  instances,  they  were  subject  to  regular  progressive 
changes,  which  seemed  to  indicate  a  real  motion  of  the  stars 
with  respect  to  each  other.  After  continuing  his  observations 
for  a  period  of  twenty -five  years,  he  satisfactorily  ascertained 
that  the  changes  in  question  were  in  reality  produced  by  a  mo- 
tion  of  revolution  of  one  star  around  the  other,  or  of  both 


CLUSTERS    OF    STARS — XEBUL.K.  209 

around  their  common  centre  of  gravity  ;  and  in  two  papers, 
published  in  the  Philosophical  Transactions  for  the  years  1803 
and  1804,  he  announced  the  important  discovery  that  there 
exist  sidereal  systems,  composed  of  two  stars  revolving  about 
each  other  in  regular  orbits.  These  stars  have  received  the 
appellation  of  Binary  Stars,  to  distinguish  them  from  other 
double  stars  which  are  not  thus  physically  connected,  and 
whose  apparent  proximity  may  be  occasioned  by  the  circum- 
stance of  their  being  situated  on  nearly  the  same  line  of  direc- 
tion from  the  earth,  though  at  very  different  distances  from  it. 

527.  Since  the  time  of  Sir  W.  Herschel,.  the  observations  upon 
the  binary  stars  have  been  continued  by  several  distinguished 
astronomers.  From  the  observations  made  upon  some  of  them, 
astronomers  have  been  enabled  to  deduce  the  form  of  their  orbits,, 
and  approximately  the  lengths  of  their  periods.  The  orbits  are 
ellipses  of  considerable  eccentricity.  The  periods  are  of  various 
lengths,  as  will  be  seen  from  the  following  enumeration  of  those 
which  are  considered  as  the  best  ascertained :  y  Leonis  1200  years; 
7  Virginis  629  years ;  61  Cygni  452  years  ;  tf  Coronas  287  years  ; 
Castor  253  years  ;  70  Opiuchi  80  years  ;  |  Ursas  58  years  ;  ^  Can- 
cri  55  years  ;  and  y\  Coronse  43  years. 

Clusters  of  Stars — Nehulce. 

528.  Many  spaces  are  discovered  in  the  heavens  which  are 
faintly  luminous,  and  shine  with  a  pale  white  light.  These  are 
called  Nehulce.  Some  are  visible  to  the  naked  eye,  but  the  great- 
er number  cannot  be  seen  without  the  aid  of  a  good  telescope. 
On  applying  to  them  telescopes  of  great  power,  they  are  found  for 
the  most  part  to  consist  of  a  multitude  of  small  stars,  distinctly 
separate,  but  very  near  each  other,  and  more  or  less  condensed 
towards  the  centre. 

529.  There  are  also  clusters  of  stars  in  close  proximity,  dis- 
persed here  and  there  over  the  sphere  of  the  heavens,  which  are 
seen  to  be  such  with  the  naked  eye,  or  with  telescopes  of  only 
moderate  power.  One  of  the  most  conspicuous  of  these  clusters, 
is  that  called  the  Pleiades. 

To  the  unaided  sight  it  appears  to  consist  of  six  or  seven  stars, 
but  a  telescope  even  of  moderate  power  exhibits  within  the  space 
they  occupy  fifty  or  sixty  conspicuous  stars.     The  constellation 
27 


210  ASTRONOMY. 

called  Coma  Berenices^  is  another  group,  more  diffused,  and  com- 
posed of  larger  stars. 

In  the  constellation  Cofwcer  there  is  a  luminous  spot,  or  nebula, 
called  PrcRsepp.,  or  the  bee-hive,  which  a  telescope  of  moderate 
power  resolves  entirely  into  stars.  In  Perseus  is  another  spot 
crowded  with  stars,  which  become  separately  visible  with  a  good 
telescope. 

530.  A  considerable  number  of  nebulae  are  met  with  in  differ- 
ent parts  of  the  heavens,  which  offer  no  appearance  of  stars,  even 
when  examined  with  telescopes  of  the  highest  power.  A  very 
great  diversity  of  form  and  aspect  obtains  among  them.  One 
of  the  most  prominent  is  that  near  the  star  v  in  Andromeda. 
It  is  visible  to  the  naked  eye,  and  has  often  been  mistaken  for  a 
comet. 


PART   III. 

OF  THE  THEORY  OF  UNIVERSAL  GRAVITATION. 


CHAPTER     XX. 

OF    THE    PRINCIPLE    OF    UNIVERSAL    GRAVITATION. 

531.  It  is  demonstrated  in  treatises  on  Mechanics,  that  if  a  body- 
move  in  a  curve  in  such  a  manner  that  the  areas  traced  by  the 
radius  vector  about  a  fixed  point,  increase  proportionally  to  the 
times,  it  is  solicited  by  an  incessant  force  constantly  directed  to- 
wards this  point.  Now,  by  Kepler's  first  law,  the  areas  described 
by  the  radii  vectores  of  the  planets  about  the  sun,  are  proportion- 
al to  the  times.  It  follows  therefore  from  this  law,  that  each 
planet  is  acted  upon  by  a  force  which  urges  it  continually  to- 
wards the  sun. 

This  fact  is  technically  expressed,  by  saying-  that  the  planets 
gravitate  towards  the  sun,  and  the  force  which  urges  each  planet 
towards  the  sun  is  called  its  Gravity,  or  Force  of  Gravity  towards 
the  sun. 

532.  It  is  also  proved  by  the  principles  of  Mechanics,  that  if  a 
body,  continually  urged  by  a  force  directed  to  some  point, 
describe  an  ellipse  of  which  that  point  is  a  focus,  the  force  by 
which  it  is  urged  must  vary  inversely  as  the  square  of  the  dis- 
tance. It  therefore  follows  from  Kepler's  second  law,  viz  :  that 
the  planets  describe  ellipses  having  the  centre  of  the  sun  at  one 
of  their  foci ;  that  the  force  of  gravity  of  each  planet  towards  the 
sun,  varies  inversely  as  the  square  of  the  distance  from  the  sun's 
centre. 

533.  By  taking  into  view  Kepler's  third  law,  it  is  proved  that  it 
is  one  and  the  same  force,  modified  only  by  distance  from  the 


212  ASTRONOMY. 

sun,  which  causes  all  the  planets  to  gravitate  towards  him,  and 
retains  them  in  their  orbits.  This  force  is  conceived  to  be  an 
attraction  of  the  matter  of  the  sun  for  the  matter  of  the  planets, 
and  is  called  the  ^olar  Attraction. 

534.  The  motions  of  the  satellites  are  in  conformity  with  Kep- 
ler's laws  ;  hence,  the  planets  which  have  satellites  are  endowed 
with  an  attractive  force  of  the  same  nature  with  that  of  the  sun. 

535.  The  existence  of  a  similar  attractive  power  in  each  of  the 
planets  that  are  devoid  of  satellites,  is  proved  by  the  fact  that  the 
observed  inequalities  of  their  motions,  and  of  those  of  the  other 
planets,  maybe  shown  upon  this  supposition  to  be  necessary  con- 
sequences of  the  attractions  of  the  planets  for  each  other. 

536.  In  like  manner  the  inequalities  in  the  motions  of  the 
satellites  and  their  primaries,  show  that  the  satellites  possess  the 
same  property  of  attraction  as  the  sun. 

537.  We  learn  from  the  motions  produced  by  the  action  of 
the  sun  and  planets  upon  each  other,  that  the  intensities  of  their 
attractive  forces  are,  at  the  same  distance,  proportional  to  their 
masses,  and  that  the  whole  attraction  of  the  same  body  for  different 
bodies,  is,  at  the  same  distance,  proportional  to  the  masses  of  these 
bodies.  From  which  we  may  infer  that  a  mutual  attraction  ex- 
ists between  the  particles  of  bodies,  and  that  the  whole  force  of 
attraction  of  one  body  for  another,  is  the  result  of  the  attractions 
of  its  individual  particles.  Moreover,  analysis  shows,  that  in 
order  that  the  law  of  attraction  of  the  whole  body  may  be  that  of 
the  inverse  ratio  of  the  square  of  the  distance,  this  must  also  be 
the  law  of  attraction  of  the  particles.  The  fact,  as  well  as  the 
law  of  the  mutual  attraction  of  particles,  is  also  revealed  by  the 
tides,  and  other  phenomena  referable  to  such  attraction. 

538.  The  celestial  phenomena  compared  with  the  general 
laws  of  motion,  conduct  us  therefore  to  this  great  principle  of 
nature  ;  namely,  that  all  particles  of  matter  inntiially  attract 
each  other  in  the  direct  ratio  of  their  7?iasses,  atid  in  the  inverse 
ratio  of  the  squares  of  their  distances.  This  is  called  the  prin- 
ciple of  Universal  Gravitation.  The  theory  of  its  existence  was 
first  promulgated  by  Sir  Isaac  Newton,  and  is  hence  often  called 
Newton^ s  Theory  of  Universal  Gravitation.  The  force  which 
urges  the  particles  of  matter  towards  each  other,  is  called  the 
Force  of  Gravitation,  or  the  Attraction  of  Gravitatio7i. 


THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE   PLANETS.       213 

539.  In  the  following  chapters  our  object  will  be  to  develope 
the  most  important  effects  of  the  principle  of  gravitation  thus 
arrived  at  by  induction.  The  perfect  accordance  that  will  be 
observed  to  obtain  between  the  deductions  from  the  theory  of 
universal  gravitation  and  the  results  of  observation,  will  afford 
additional  confirmation  of  the  truth  of  the  theory. 


CHAPTER    XXI. 


THEORY    OF    THE    ELLIPTIC    MOTION    OF    THE    PLANETS. 

540.  Let  the  attraction  of  the  unit  of  mass  of  the  sun  for  the 
unit  mass  of  a  planet,  at  the  unit  of  distance,  be  designated  by  1. 
The  whole  attraction  exerted  by  the  sun  upon  the  unit  of  mass, 
at  the  same  distance,  will  then  be  expressed  by  the  mass  of  the  sun 
(M) ;  or,  in  other  words,  by  the  number  of  units  which  its  mass 
contains.     And  the  attraction  F,  at  any  distance  r,  will  result 

M 

from  the  proportion  M  :  F  :  :  r^  :  1-,  which  gives  F  =  — .  This, 

in  the  language  of  Dynamics,  is  the  Accelerating  Force  soliciting 
the  planet. 

As    —  expresses  the  attraction  of  the  sun  for  a  unit  of  mass  of 

the  planet,  its  attraction  for  the  entire  mass  m  of  the  planet  will 

M 
be  expressed  by  m  —     This  is  the  moving  force  of  the  planet, 
r- 

and  since  it  is,  at  the  same  distance,  proportional  to  the  mass  of 

the  planet,  the  velocity  due  to  its  action  is  the  same,  whatever 

may  be  the  mass. 

541.  The  planet  has  also  an  attraction  for  the  sun,  as  well  as 
the  sun  for  the  planet,  and  the  expression  for  its  attractive  force, 
or  for  the  accelerating  force  animating  the  sun,  Mali  obviously 


214  ASTRONOMY. 

be   —     The  sun  will  then  tend  towards  the  planet,  as  the  planet 

r-. 
towards  the  sun.  But,  if  the  two  bodies  were  to  set  out  from 
a  state  of  rest,  the  velocity  of  the  planet  would  be  as  many- 
times  greater  than  the  velocity  of  the  sun,  as  the  mass  of  the  sun  is 
greater  than  that  of  the  planet.  For,  the  velocity  of  the  planet 
would  be  to  that  of  the  sun  as  the  attractive  force  of  the  sun  is  to 

M     thj 
the  attractive  force  of  the  planet,  that  is,  as  —  :  — ,  or  as  M :  m. 

r^      r- 

As  the  attraction  of  the  particles  of  the  sun  and  planet  are  mu- 
tual and  equal,  the  attraction  of  the  planet  for  the  entire  mass  of 
the  sun  must  be  equal  to  the  attraction  of  the  sun  for  the  entire 
mass  of  the  planet. 

542.  The  sun  and  any  flanet  revolve  about  their  common 
centre  of  gravity. 

To  show  this,  we  would  remark,  in  the  first  place,  that  it  is  a 
principle  of  Mechanics  that  the  mutual  actions  of  the  different 
members  of  a  system  of  bodies  cannot  affect  the  state  of  the  cen- 
tre of  gravity  of  the  system.  This  is  called  the  Principle  of  the 
Preservation  of  the  Centre  of  Gravity.  It  follotvs  from  it  that  the 
common  centre  of  gravity  of  the  sun  and  any  planet  is  at  rest, 
unless  it  has  a  motion  of  translation  in  common  with  the  two 
bodies,  imparted  by  a  force  extraneous  to  the  system.  As  we  are 
concerned  at  present  only  with  the  relative  motion  of  the  sun  and 
planet,  such  motion  of  translation,  if  it  does  exist,  may  be  left  out  of 
account.  Now,  let  S  (Fig.  71)  be  the  sun,  and  P  any  planet,  sup- 
posed for  the  moment  to  be  at  rest.  If  neither  of  the  two  bodies 
should  receive  a  velocity  in  a  direction  oblique  to  P  S  the  line  of 
their  centres,  they  would  move  towards  each  other  by  virtue  of 
their  mutual  attraction,  and  meet  at  C  their  common  centre  of 
gravity.*  But,  if  the  body  P  have  a  projectile  velocity  given 
to  it  in  any  direction  P  t,  inclined  to  the  line  PS,  it  is  susceptible 
of  proof  that  its  motion  relative  to  the  sun  may  be  in  an  ellipse, 
as  is  observed  to  be  the  case  with  the  planets. 

Now,  while  the  planet  moves  in  space,  the  line  of  the  centres 


*  The  common  centre  of  gravity  of  two  bodies  lies  on  the  line  joining  their 
centres,  and  divides  this  line  into  parts  inversely  proportional  to  the  masses  of 
the  bodies. 


THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS.       215 

of  the  planet  and  snn  must  continually  pass  through  the  station- 
ary position  of  the  centre  of  gravity,  and  therefore,  when  the 
planet  has  advanced  to  any  point  p,  the  sun  will  have  shifted  its 
position  to  some  point  s  on  the  line  p  C  prolonged.  Moreover,  as 
the  two  bodies  mutually  gravitate  towards  each  other,  the  paths 
of  each  in  space  will  be  continually  e^eefee^^  towards  the  other  u^/-^***^ 
body,  and,  therefore,  also  towards  the  centre  of  gravity  C,  which 
is  constantly  in  the  same  direction  as  the  other  body.  Since  the 
planet  performs  a  revolution  around  the  sun,  the  sun  and  planet 
must  each  continue  to  move  about  the  point  C  until  they  have 
accomplished  a  revolution  and  returned  to  the  line  PCS.  Also 
as  the  distance  P  S  of  the  two  bodies  will  be  the  same  at  the  end 
as  at  the  beginning  of  the  revolution,  as  well  as  the  ratio  of  their 
distances  P  C  and  S  C  from  the  centre  of  gravity,  they  will  re- 
turn to  the  positions  PS,  from  which  they  set  out,  and  will 
therefore  move  in  continuous  curves. 

As  the  distances  of  the  sun  and  planet  from  their  common 
centre  of  gravity  are  constantly  reciprocally  proportional  to  their 
masses,  the  orbit  of  the  sun  will  be  exceedingly  small  in  com- 
parison with  the  orbit  of  the  planet. 

543.  If  to  both  the  sun  and  planet  there  should  be  applied  a 

force  equal  to  the  accelerating  force  of  the  sun,  —  (Art.  541),  but 

in  an  opposite  direction,  the  sun  would  be  solicited  by  two  forces 
that  would  destroy  each  other,  but  the  planet  would  now  be 
urged  towards  the  sun  remaining  stationar3^  with  the  accelerating 

force       "^  ^,  or  a  force  the  intensity  of  wliich  was  equal  to  the 

J.  2 

sum  of  the  intensities  of  the  attractive  forces  of  the  sun  and  planet, 
at  the  distance  of  the  planet.  Now,  the  application  of  a  common 
force  will  not  alter  the  relative  motion  of  the  two  bodies.  Hence, 
in  investigating  this  motion  we  are  at  liberty  to  conceive  the  sun 
to  be  stationary,  if  we  suppose  the  planet  to  be  solicited  by  the 

accelerating   force  51+^.     As  the  mass  of  the  sun  is  very  much 

greater  than  that  of  any  planet,  but  little  error  will  be  committed 
in  neglecting  the  attraction  of  the  planet,  and  taking  into  account 

only  the  sun's  action  — 


216  ASTUU.NOMV. 

544.  Analysis  makes  known  the  general  laws  of  the  motion 
of  a  body,  when  impelled  by  a  projectile  force,,  and  afterwards 
continually  attracted  towards  the  sun's  centre  by  a  force  vary- 
ing inversely  as  the  square  of  the  distance.  We  learn  by  it 
that  the  body  will  necessarily  describe  some  one  of  the  conic 
sections,  around  the  sun  situated  at  one  of  its  foci.  We  learn, 
also,  that  the  nature  of  the  orbit,  as  well  as  the  length  of  the 
major  axis,  is  wholly  dependent,  for  any  given  distance  of  the 
planet,  upon  the  intensity  of  the  projectile  force,  but  that  the 
position  of  the  axis,  and  the  eccentricity  of  the  orbit,  depend 
also  upon  the  angle  of  projection,  (that  is,  the  angle  included, 
at  the  commencement  of  the  motion,  between  the  line  of  direc- 
tion of  the  projectile  force  and  the  radius  vector.)  As  to  the 
relative  intensity  of  the  projectile  force  necessary  to  the  produc- 
tion of  each  one  of  the  conic  sections,  a  certain  intensity  of 
force  will  produce  a  parabola;  any  less  intensity,  an  ellipse  or 
circle  ;  and  any  greater,  a  hyperbola. 

545.  If  the  velocity  that  would  at  a  given  distance  be  im- 
parted by  the  sun's  attraction  in  a  second  of  time,  which  is  the 
measure  of  its  intensity  at  the  given  distance,  be  found,  and  also 
the  distance  of  a  planet  at  any  time,  as  well  as  its  velocity  and 
the  angle  made  by  the  direction  of  its  motion  with  the  radius 
vector,  the  form,  dimensions,  and  position  of  the  planet's  orbit 
can  be  computed.  This  is  to  determine  the  orbit  a  priori. 
The  practice  has  been,  however,  to  determine  the  various 
elements  of  a  planet's  orbit  by  observation,  (as  already  described. 
Chap.  VIII). 

The  elements  being  known,  the  equations  of  the  elliptic  mo- 
tion, investigated  on  the  principles  of  Mechanics,  serve  to  make 
known  the  position  and  velocity  of  the  planet  at  any  time. 
(The  investigation  of  these  equations  may  be  found  in  the  En- 
cyclopaedia Metropolitana,  Article  Physical  Astronomy,  page 
653,  in  the  Mecanique  Elementaire  de  Francoeur,  and  in 
many  other  similar  works.)* 

546.  The  physical  theory  of  the  motion  of  a  satellite  around 


*  The  equations  arc  the  same  with  those  deduced  directly  from  Kepler's  laws  of 
the  planetary  motions.     (See  App.,  Solution  of  Kepler's  Problem.) 


THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS.       217 

its  primary  is  obviously  the  same  as  that  of  the  motion  of  a 
planet  around  the  sun. 

547.  According  to  the  principle  of  the  preservation  of  the 
centre  of  gravity  (Art.  542),  the  centre  of  gravity  of  the  whole 
solar  system  must  either  be  at  rest,  or  have  a  motion  of  transla- 
tion in  space  in  common  with  the  system,  resulting  from  the 
action  of  a  foreign  force.  If  any  general  motion  of  the  solar 
system  subsists,  it  has  not  yet  been  recognized  from  obser- 
vation. 

548.  The  sun  and  planets  revolve  around  their  common 
centre  of  gravity.  The  path  of  the  sun's  centre  results  from 
the  joint  action  of  all  the  planets,  and  is  a  complicated  curve. 
As  the  quantity  of  matter  in  all  the  planets  taken  together  is 
very  small,  compared  with  that  in  the  sun,  the  extent  of  the 
curve  described  by  the  centre  of  the  sun  cannot  be  very  great. 
It  is  found  by  computation,  that  the  distance  between  the  sun's 
centre  and  the  centre  of  gravity  of  the  system  can  never  be 
equal  to  the  sun's  diameter. 

549.  It  is  demonstrated  in  treatises  on  Mechanics,  that  if 
foreign  forces  act  upon  a  system  of  bodies,  the  centre  of  gravity 
of  the  system  will  move  just  as  the  whole  mass  of  the  system 
concentrated  at  the  centre  of  gravity  would  move,  under  the 
action  of  the  same  forces.  It  follows  from  this  principle,  that 
from  the  attraction  of  the  sun  for  a  primary  planet  and  its  satel- 
lites, their  common  centre  of  gravity  will  revolve  around  the 
sun,  just  as  the  whole  quantity  of  matter  in  the  planet  and  its 
satellites  concentrated  at  this  point  would,  under  the  influence 
of  the  same  attraction.  Moreover,  the  same  considerations 
which  show  that  the  sun  and  planets  revolve  about  their  com- 
mon centre  of  gravity,  will  also  show  that  a  primary  planet  and 
its  satellites  revolve  about  their  common  centre  of  gravity.  It 
appears,  therelbre,  that  in  the  case  of  a  planet  which  has  satel- 
lites, it  is  not,  strictly  speaking,  the  centre  of  the  planet  that 
moves  agreeably  to  the  first  and  second  laws  of  Kepler,  but  the 
common  centre  of  gravity  of  the  planet  and  its  satellites  ;  the 
planet  and  satellites  revolving  around  the  centre  of  gravity,  as 
it  describes  its  orbit  about  the  sun. 

550.  It  may  be  worth  while  here  to  remark,  that  the  revolu- 
tion of  the  earth  around  the  common  centre  of  gravity  of  the 

28 


218  ASTRONOMY. 

earth  and  moon,  occasions  an  inequality,  both  of  longitude  and 
latitude,  in  the  apparent  motion  of  the  sun.  It  is,  however,  ex- 
ceedingly small,  for  the  reason  that  the  distance  of  the  earth's 
centre  from  the  centre  of  gravity  is  very  short,  in  comparison 
with  the  distance  of  the  sun.  The  mass  of  the  earth  is  to  that 
of  the  moon  as  80  to  1,  while  the  distance  of  the  moon  is  to  the 
radius  of  the  earth  as  60  to  1.  It  follows,  therefore,  that  the 
common  centre  of  gravity  of  the  earth  and  moon  lies  within  the 
body  of  the  earth. 

551.  It  appears  also  from  the  physical  investigation  of  the 
elliptic  motion  of  the  planets,  that  Kepler's  third  law  is  not  rigor- 
ously true.  In  consequence  of  the  action  of  the  planets  upon 
the  sun,  the  ratio  of  the  periodic  times  of  the  different  planets  de- 
pends upon  the  masses  of  the  planets,  as  well  as  their  distances 
from  the  sun.  If  y?  and  j)'  be  the  periodic  times  of  any  two  of  the 
planets,  a  and  a'  their  mean  distances  from  the  sun's  centre,  and 
771  and  m'  their  quantities  of  matter,  that  of  the  sun  being  denoted 
by  1,  then,  disregarding  the  actions  of  the  other  planets, 

p2    .  pl2    ■    ■ 


\-\-m     1  +  w' 

As  m  and  m'  are  very  small  fractions,  the  error  resulting  from 
their  omission  will  be  very  small.  If  we  omit  them,  we  shall 
have, 


J)-  :  ]j'^  :  :  a^  .a' 


which  is  Kepler's  third  law. 


CHAPTER    XXII. 

THEORY    OF    THE    PERTURBATIONS    OF    THE    ELLIPTIC    MOTION 
OF    THE    PLANETS    AND    OF    THE    MOON. 

552.  We  have,  in  a  previous  chapter,  given  a  general  idea  of 
the  mode  of  determining  from  theory  and  observation  combined, 
the  law  and  amount  of  the  perturbations  or  inequalities  of  the 


THEORY    OF    PERTURBATIONS — DISTURBING    FORCES."   '219 

lunar  and  planetary  motions.  We  propose  now  to  give  some  in- 
sight into  the  nature  and  manner  of  operation  of  the  disturbing 
forces,  and  will  commence  with  the  perturbations  of  the  moon 
produced  by  the  action  of  the  sun. 

553.  We  have  already  (Art.  259)  showed  how  the  intensity  and 
direction  of  the  disturbing  force  of  the  sun,  in  any  given  position 
of  the  moon  in  its  orbit,  may  be  determined.  Let  us  now  derive 
the  disturbing  forces  that  take  effect  in  the  three  directions  in 
which  the  motion  of  the  moon  can  be  changed  ;  namely,  in  the 
direction  of  the  radius  vector,  of  the  tangent  to  the  orbit,  and  of 
the  perpendicular  to  its  plane.  Let  E  (Fig.  72)  be  the  earth,  M 
the  moon,  and  S  the  sun.  Let  the  force  exerted  by  the  sun  upon 
the  moon  be  decomposed  into  two  forces,  one  acting  along  the 
line  M  S'  parallel  to  E  S,  and  the  other  from  M  towards  E.  If 
the  component  along  M  S'  were  equal  to  the  force  exerted  by  the 
sun  upon  the  earth,  the  motion  of  the  moon  about  the  earth  would 
not  be  changed  by  the  action  of  these  two  forces.  Hence,  the 
difference  between  them  will  be  the  disturbing  force  in  the  direc- 
tion M  S'.  The  component  along  M  E  is  another  disturbing 
force.  It  is  called  the  Addititious  Force,  because  it  tends  to  in- 
crease the  gravity  of  the  moon  towards  the  earth.  The  disturb- 
ing force  along  MS'  will  generally  be  inclined  to  the  plane  of  the 
orbit,  and  may  be  decomposed  into  three  forces,  one  in  the  direc- 
tion of  the  tangent,  another  in  the  direction  of  the  radius  vector, 
and  a  third  in  the  direction  of  the  perpendicular  to  the  plane. 
The  first  mentioned  component  is  called  the  Tangential  Force  ; 
the  second  is  called  the  Ablatitioiis  Force  ;  and  the  third  we  shall 
call  the  Perpendicular  Force. 

The  actual  disturbing  force  in  the  direction  of  the  radius  vector 
is  'equal  to  the  difference  between  the  addititious  and  ablatitious 
forces,  and  is  called  the  Radial  Force.  This  and  the  tangential 
and  perpendicular  forces  constitute  the  disturbing  forces,  the  di- 
rect operation  of  which  is  to  be  considered. 

554.  To  obtain  general  analytical  expressions  for  these  forces, 
let  the  distance  of  the  sun  from  the  earth  (which  for  the  present 
we  shall  suppose  to  be  constant)  be  denoted  by  a,  and  the  dis- 
tances of  the  moon  from  the  earth  and  sun,  respectively,  by  y,  and 
z.  Also  let  F  =  the  force  exerted  by  the  earth  upon  the  moon, 
P  =  the  force  exerted  by  the  sun  upon  the  earth,  and  Q,  =  the  force 


220  ASTRONOMY. 

exerted  by  the  sun  upon  the  moon.  Then,  if  we  denote  the  mass 
of  the  earth  by  1,  and  take  m  to  stand  for  the  mass  of  the  sun,  we 
shall  have,  ( Art.  539), 

2/2  a2  z^ 

Let  the  force  d  be  represented  by  the  line  M  S  (Fig.  72) ;  and 
let  its  component  parallel  to  E  S,  or  M  S'  =  R,  and  its  component 
along  the  radius  vector,  or  M  E  =  T. 

a:T  :  :MS  :ME;  or,    !^  :  T::2;:y. 

z- 

Whence,  addititious  force  T  =  — ^  .  .  .  (130). 

In  a  similar  manner  we  obtain, 

R  =  ^^?^  .  .  .  (131). 

The  disturbing  force  in  the  direction  of  the  sun 


T,       ^      m  a       in 
=  R  —  P  = —  —  -  in  a 


(-    — V 

\z^  a^  f 


Z''       a^ 

Now,  let  a,  /3,  y,  denote  the  angles  made  by  the  line  M  S', 
respectively,  with  the  tangent,  the  radius  vector,  and  the  perpen- 
dicular to  the  plane  of  the  orbit,  and  we  shall  have  for  the  compo- 
nents of  the  disturbing  force  R  —  P,  along  these  lines  ; 

tangential  force  —m  a  i — 1  cos  a  .  .  .  (132) ; 

ablatitious  force  —  mai — I  cos  (3  .  .  .  (133) ; 

y  z^         a^    ' 

perpendicular  force  =  in  a\ — I  cos  y  .  •  .  (134). 

Combining  equation  (133)  with    equation  (130)  we  obtain  for 
the  radial  force, 


radial  force  =  m  y  —  mai — I 

^     z^  \z  ""        a^   r 


cos 


r- 


555.  The  obliquity  of  the  orbit  of  the  moon  to  the  plane  of 
the  ecliptic,  affects  but  very  slightly  the  value  of  the  tangential 
and  radial  forces.  If  we  leave  it  out  of  account,  or  suppose  the 
moon's  orbit  to  lie  in  the  plane  of  the  ecliptic,  we  shall  have 


INVESTIGATION    OF    THE    DISTURBING    FORCES.  221 

(Fig.  73)  /3  =  S'  M  L  =  S  E  M  the  elongation  of  the  moon  =  tp, 
and  a  —  complement  of  cp,  which  gives, 

tangential  force  =  nia  i ■  sin  9  .  .  .  (135). 

radial  force  —  my —  m a  I I  cos (p  .  .  .  (136). 

556.  Equation  (134)  may  be  transformed  into  another,  which 
is  better  adapted  to  the  purposes  we  have  in  view.  Let  M  K 
(Fig.  72)  represent  the  perpendicular  to  the  plane  of  the  moon's 
orbit,  M  F  the  intersection  of  the  plane  S  M  K  with  the  plane  of 
the  moon's  orbit,  and  S  I,  I  F  the  intersections  of  a  plane  pass- 
ing through  S  and  perpendicular  to  E  N,  the  line  of  nodes, 
with  the  plane  of  the  ecliptic  and  the  plane  of  the  orbit.  S  F 
will  be  perpendicular  to  both  I  F  and  M  F.  Denote  S  I  F  the 
inclination  of  the  orbit  to  the  ecliptic,  by  I,  S  E  N  the  angular 
distance  of  the  sun  from  the  node  by  N,  and  S  E  and  S  M  by  a, 
and  2r,  as  before. 

Now,  in  equation  (134)  7  stands  for  the  angle  S'MK,  but  S' 
M  K  -  S  M  K  (nearly),  and, 

cos  S  M  K  =  sin  S  M  F  =  |Z ; 

S  F  =  S  Isin  S  I F,  and  S  I  =  S  E  sin  S  E  I ; 
whence,         S  F  -■  S  E  sin  S  E  I  sin  S  I  F  =  a  sin  N  sin  I ; 

substituting, 

o  T\T  ir      «  sin  N  sin  I      a  sin  N  sin  I 
cos  7  =  cos  S  M  K  = — — = 

Thus  we  have, 

r                    /I          1  \a  sin  N  sin  I  /io.r\ 

perpen.  iorce  =  m  a  \  —  —  — I .  .  .  (137). 

\  z"^       a^f  z 

557.  The  variable  z  may  be  eliminated  from  equations  (135), 
(136)  and  (137),  and  other  equations  obtained,  involving  only 
the  variables  y  and  (p.  Let  M  L  (Fig.  72)  be  drawn  through 
the  place  of  the  moon  perpendicular  to  E  S.  Then,  using  the 
same  notation  as  in  the  preceding  articles, 

L  S  -  2r  (nearly),  E  L  -  E  M  cos  L  E  M  =  y  cos  9  ; 
but,  LS=SE  —  EL; 

whence,       z^  a  —  y  cos  9,  and  z"^  =a^  —  3a^  y  cos  0  : 


222  ASTRONOMY. 

neglecting  the  terms  containing  the  higher  powers  of  y  than 
the  first,  as  they  are  very  minute,  y  being  only  about  ^\^  a. 

1  _  1 ^  J_  ,  3ycos(p  . 

^     a"^  — 3  a'y  cos(p      a^         a* 

neo-lectino-  all  the  terms  of  the  quotient   that  involve  higher 

1 
powers  of  y  than  the  first.     Substituting  this  value  of  —  m 

equation  (135),  we  obtain, 

.  ,  r           3  m  y  cos  (p  sin  0  . 
taDffential  lorce  = ^ , 

or,  (App.  For.  13), 

tangential  force  = ^—^ .  .  .  (138). 

Making  the  same  substitution  in  equation  (136),  and  neglecting 
the  term  containing  y-,  there  results, 

J-  1  r             my  (I  —  3  cos2<p) 
radial  force  =  — iLS 1^ 

or,  (App.  For.  9), 

radial  force  =  -  ^ya  +  3cos2^  ^^3^^ 

In  equation  (137)  we  have  to  substitute,  besides,  the  value  of 
z,  viz  :  a  —  y  cos  9  ;  then  dividing  and  neglecting  as  before,  we 
have, 

perpen.  force  =  ?^^-|^!^  sin  N  sin  I  .  .  .  (140.) 

558.  If  the  disturbing  forces  retained  constantly  the  same 
intensity  and  direction,  the  result  would  be  a  continual  pro- 
gressive departure  from  the  elliptic  place  ;  but,  in  point  of 
fact,  these  forces  are  subject  to  periodical  changes  of  intensity 
and  direction  from  several  causes,  from  which  results  a  com- 
pensation of  effects,  and  an  eventual  return  to  the  elliptic 
place.  The  causes  of  the  variation  of  the  disturbing  forces 
are  : 

1.  The  revolution  of  the  moon  around  the  earth. 

2.  The  elliptic  form  of  the  apparent  orbit  of  the  sun. 

3.  The  elliptic  form  of  the  orbit  of  the  moon. 

4.  The  inclination  of  the  two  orbits. 


VARIATION    OF    THE    TANGENTIAL    FORCE.  223 

As  the  variations  of  the  radial  and  tangential  forces,  resulting 
from  the  inclination  of  the  orbits,  are  very  minute,  we  shall 
leave  them  out  of  account,  and  in  the  consideration  of  the 
effects  of  these  forces,  shall,  for  the  sake  of  simplicity,  regard 
the  orbits  as  lying  in  the  same  plane. 

The  first  mentioned  circumstance  is  the  most  prominent 
cause  of  variation,  and  gives  rise  to  the  more  conspicuous 
perturbations.  The  other  two  serve  to  modify  the  variations 
of  the  forces  resulting  from  the  first,  and  occasion  each  a  dis- 
tinct set  of  periodical  perturbations. 

559.  Let  us  now  investigate,  in  succession,  the  effects  of  each 
of  the  disturbing  forces,  commencing  with  the  tangential  force. 
The  tangential  force  takes  effect  directly  upon  the  velocity  of  the 
moon  in  its  orbit ;  and  as  its  line  of  direction  does  not  pass 
through  the  earth,  it  disturbs  the  equable  description  of  areas.  It 
also  affects  the  radius  vector  indirectly,  by  changing  the  centrifu- 
gal force.  To  understand  the  detail  of  its  action,  we  must  in- 
quire into  the  variations  which  it  undergoes. 

If  we  regard  y  as  constant  in  the  expression  for  the  tangential 
force,  (equa.  138),  which  amounts  to  considering  the  moon's 
orbit  as  circular,  the  expression  will  become  equal  to  zero  when 
sin  2  (p  =  0,  and  will  have  its  maximum  value  when  sin  2  (p  =  1, 
It  will  also  change  its  sign  with  sin  2  9.  It  appears,  therefore, 
that  the  tangential  force  is  zero  in  the  syzigies  and  quadratures, 
where  it  also  changes  its  direction,  and  that  it  attains  its  maxi- 
mum  value  in  the  octants.  It  Avill  be  seen,  on  inspecting  Fig, 
74,  that  it  will  be  a  retarding  force  in  the  first  quadrant,  (A  B). 
Accordingly,  it  will  be  an  accelerating  force  in  the  second,  a 
retardinof  force  agfain  in  tJie  third,  and  an  accelerating  force 
again  in  the  fourth. 

This  will  also  appear  upon  considering  the  direction  of  the 
disturbing  force  parallel  to  the  line  of  the  centres  of  the  sun 
and  earth,  in  the  various  quadrants.  In  the  nearer  half  of  the 
orbit  the  sun  tends  to  draw  the  moon  away  from  the  earth,  and 
the  force  in  question  is  directed  towards  the  sun.  In  the  more 
remote  half  the  sun  tends  to  draw  the  earth  away  from  the 
moon,  but  we  may  regard  it,  instead,  as  urging  the  moon  from 
the  earth  by  the  same  force  ;  for,  the  relative  motion  will  be  the 
same  on  this  supposition.     In  the  part  of  the  orbit  supposed, 


224  ASTRONOMY. 

then,  the  disturbing  force  under  consideration  will  be  directed 
from  the  sun,  as  represented  in  Fig.  74. 

560.  It  appears,  then,  that  the  tangential  force  will  alternately 
retard  and  accelerate  the  motion  of  the  moon  during  its  passage 
through  the  different  quadrants,  and  that  the  maximum  of  velocity- 
will  occur  in  the  syzigies  A,  C,  where  the  accelerating  force  be- 
comes zero,  and  the  minimum  of  velocity  in  the  quadratures  B,D, 
where  the  retarding  force  becomes  zero.  On  the  supposition  that 
the  orbit  is  a  circle,  the  arcs  A  B,  EC,  CD,  and  D  A,  would  be 
equal,  and  the  retardation  of  the  velocity  in  one  quadrant  would 
be  compensated  for  by  an  equal  acceleration  in  the  next,  and  at 
the  close  of  a  synodic  revolution,  the  velocity  of  the  moon  would 
be  the  same  as  at  its  commencement.  As  the  velocity  is  greatest 
in  the  syzigies,  and  least  in  the  quadratures,  and  as  the  degree  of 
retardation  is  the  same  as  that  of  acceleration,  the  mean  motion* 
must  have  place  in  the  octants.  Now,  as  the  moon  moves  from 
the  syzigy  A  with  a  motion  greater  than  the  mean  motion,  her  true 
place  will  be  in  advance  of  her  mean  place,  and  will  become  more 
and  more  so  till  she  reaches  the  octant,  where  the  true  motion  is 
equal  to  the  mean.  The  difference  between  the  true  and  mean 
place  will  then  be  the  greatest ;  for  after  that,  the  true  motion  be- 
coming less  than  the  mean,  the  mean  place  will  approach  nearer 
to  the  true,  till  at  the  quadrature  they  coincide.  Beyond  B,  the 
true  motion  still  contmuing  less  than  the  mean,  the  mean  place 
will  be  in  the  advance  of  the  true,  and  the  separation  will  increase 
till  at  the  octant  the  true  motion  has  attained  to  an  equality  with 
the  mean  motion,  after  which,  the  mean  motion  being  the  slowest, 
the  true  place  will  approach  the  mean  till  at  the  syzigy  C  they 
again  coincide.  Correspondmg  ellects  will  take  place  in  the  two 
remaining  quadrants.  We  perceive,  therefore,  that  the  tangential 
force  produces  an  inequality  of  longitude,  which  attains  to  its 
maximum  positive  and  negative  value  in  the  octants,  and  is  zero 
in  the  syzigies.  This  is  the  inequality  known  in  Plane  Astrono- 
my by  the  name  of  Variation,  (Art.  272). 

561.  Let  us  now  inquire  into  the  modifications  of  the  effects  of 
the  tangential  force,  that  result  from  the  elliptic  form  of  the  sun's 


*  The  expressions,  mean  motion,  true  motion,   mean   place,   true  place,   are 
hero  to  be  understood  only  in  relation  to  the  perturbation  under  consideration. 


EFFECTS    OF    THE    TANGENTIAL    FORCE.  225 

orbit.  Suppose  that  at  the  moment  when  the  moon  sets  out  from 
conjunction,  the  sun  is  in  the  apogee  of  its  orbit :  then  it  is  plain 
that,  during  the  whole  revolution  of  the  moon,  the  sun's  disturbing 
force  would  be  on  the  increase  by  reason  of  the  dimmution  of  the 
sun's  distance,  and  that,  in  consequence,  the  retardation  in  the 
first  quadrant  v/ould  be  less  than  the  acceleration  in  the  second, 
and  the  retardation  in  the  third  less  than  the  acceleration  in  the 
fourth.  So,  that,  when  the  moon  had  again  come  around  into  con- 
junction, the  acceleration  would  have  over-compensated  the  re- 
tardation. This  kind  of  action  would  go  on  so  long  as  the  sun 
approached  the  earth  :  but  when  it  had  passed  the  perigee  of  its 
orbit,  and  began  to  recede  from  the  earth,  the  reverse  effect  would 
take  place,  and  a  retardation  of  the  moon's  orbitual  motion  would 
happen  each  revolution.  If  the  anomalistic  revolution  of  the  sun 
was  an  exact  multiple  of  the  synodic  revolution  of  the  moon,  the 
acceleration  in  each  revolution  of  the  moon  during  the  passage  of 
the  sun  from  the  apogee  to  the  perigee  of  its  orbit,  would  be  com- 
pensated for  by  an  equivalent  retardation  in  the  revolution  of  the 
moon  answering  to  the  same  distance  of  the  sun  in  its  passage  from 
the  apogoo  to  the  perigee ^-and  the  velocity  of  the  moon  would  be 
the  same  at  the  close  of  an  anomalistic  revolution  of  the  sun  as  at 
its  commencement.  But  as  this  relation  does  not,  in  fact,  subsist 
between  the  anomalistic  revolution  of  the  sun  and  the  synodic 
revolution  of  the  moon,  a  compensation  between  the  accelerations 
and  retardations  answering  to  the  different  revolutions  of  the 
moon,  will  not  be  effected  until  conjunctions  shall  have  occurred 
at  every  variety  of  distance  of  the  sun  in  each  half  of  its  orbit. 
Since  the  anomalistic  and  synodic  revolutions  are  incommensura- 
ble, the  sun  will  be,  in  reality,  in  every  variety  of  position  in  its 
orbit,  at  the  time  of  conjunction,  in  process  of  time  ;  so  that  event- 
ually the  original  velocity  in  conjunction  will  be  regained.  It 
appears,  therefore,  that  the  variation  of  the  moon's  motion  from 
one  revolution  to  another,  occasioned  by  the  elliptic  form  of  the 
sun's  orbit,  is  periodic.  Its  period  will  be  the  interval  of  time  in 
which  the  moon  will  perform  a  certain  number  of  synodic  revo- 
lutions, while  the  sun  performs  a  certain  number  of  anomalistic 
revolutions.  Avoiding  unnecessary  precision,  we  find  it  to  con- 
sist of  but  a  moderate  number  of  years. 
29 


'C-^'Vaa)*' 


226  ASTRONOMY. 

562,  We  have  next  to  consider  the  consequences  of  the  ellip- 
tic form  of  the  moon's  orbit.  We  remark  in  the  first  place  that 
the  orbit  being  an  ellipse,  the  areas  A  E  B,  B  E  C,  C  E  D,  and 
D  E  A  (Fig.  74)  will  be  unequal,  and  therefore,  by  the  laws  of 
elliptic  motion,  the  arcs  A  B,  B  C,  CD,  and  D  A  will  be  de- 
scribed in  unequal  times.  It  follows  from  this,  that  the  retarda- 
tion in  the  first  quadrant  will  not  be  exactly  compensated  by 
the  acceleration  in  the  second,  and  that  the  retardation  in  the 
third  will  not  be  exactly  compensated  by  the  acceleration  in  the 
fourth.  Therefore,  at  the  end  of  the  synodic  revolution  the 
moon  will  have  an  excess  or  deficiency  of  velocity.  Its  mean 
motion  will  then  vary  from  one  revolution  to  another,  by  reason 
of  the  ellipticity  of  its  orbit.  This  variation  will  be  periodic, 
like  that  just  considered,  and  for  similar  reasons.  The  excess 
or  deficiency  of  velocity  at  the  close  of  any  one  revolution,  will 
in  time  be  compensated  by  an  equal  deficiency  or  excess  occur- 
ring at  the  close  of  another  revolution,  when  the  sun  has  a  cer- 
tain different  position  with  respect  to  the  perigee  of  the  moon's 
orbit. 

563.  We  pass  now  to  the  consideration  of  the  action  of  the 
radial  force.  The  direct  general  effect  of  the  radial  force,  is  an 
alteration  in  the  intensity  of  the  moon's  gravity  towards  the 
earth,  and  in  its  law  of  variation.  Its  specific  effects  are  period- 
ical variations  in  the  magnitude,  eccentricity  and  position  of  the 
orbit.  As  it  is  directed  towards  the  earth,  it  will  not  disturb 
the  equable  description  of  areas.  To  discover  the  variations  of 
this  force  we  have  only  to  discuss  the  general  analytical  expres- 
sion for  it,  already  investigated.     It  is, 

-,■  ^  r  my  (I  —  3  cos^  ©) 

radial  force  =  — ^-i i-L . 

a-" 

We  shall  have,  radial  force  =  0,  Avhen  1  —  3  cos^  <p  =  0,  or 
when  cos  (p  =  ±  V^.  This  value  of  cos  (p  answers  to  four  points 
lying  on  either  side  of  the  quadratures,  and  about  35°  distant 
from  them.  When  cos  9  is  numerically  greater  than  ^/i,  the 
result  will  be  negative,  and  when  it  is  less  than  -/i)  the  re- 
sult will  be  positive.  It  follows,  therefore,  that  the  radial  force 
increases  the  gravity  of  the  moon  in  the  quadratures,  and  for 
about  35°  on  each  side  of  them,  and  that  during  the  remainder 
of  a  synodic  revolution  it  diminishes  it. 


VARIATION    OF    THE    RADIAL    FORCE.  227 

When  the  moon  is  in  quadratures  cos  cp  =0,  and 
radial  force  = — ^  .  .  .  (141), 

In  the  syzigies,  we  have  cos  9  =  ±  1,  which  gives 
radial  force  = —M  .  .  .  (142). 

It  appears  then  that  the  diminution  of  the  moon's  gravity  in 
the  syzigies,  is  double  of  its  increase  in  the  quadratures. 

We  learn  also  from  equations  (141)  and  (142),  that  the  radial 
force  in  the  quadratures  and  syzigies  varies  directly  as  the  dis- 
tance;  from  which  we  conclude  that  the  gravity  of  the  moon 
varies  at  these  points  by  a  different  law  from  that  of  the  inverse 
squares.  In  the  quadratures  the  gravity  will  be  increased  most 
at  the  greatest  distance,  when  it  is  the  least :  and  thus  it  will 
vary  in  a  less  rapid  ratio  than  the  square  of  the  distance.  In 
the  syzigies  it  will  be  diminished  most  at  the  greatest  distance, 
or  when  it  is  the  least ;  and  accordingly,  at  these  points  it  will 
vary  in  a  more  rapid  ratio  than  the  square  of  the  distance. 

564.  An  easy  investigation,  with  the  aid  of  the  differential 
calculus,  proves  that  the  mean  diminution  of  the  moon's  gra- 

111  T  •  •  1  • 

vity  from  the  sun's  action  is ;  r  representing  in  this  case 

2a* 

the  mean  distance  of  the  moon  from  the  earth.     The  value  of 

this  expression  is  readily  found  to  be  equal  to  about  the  368th 

part  of  the  whole  gravity  of  the  moon  to  the  earth. 

hi  consequence  of  this  diminution,  the  moon  must  describe 

her  orbit  at  a  greater  distance  from  the  earth,  with  a  less  angfu- 

lar  velocity,  and  in  a  longer  time,  than  if  she  were  acted  on  only 

by  the  attraction  of  the  earth. 

565.  The  radial  force  of  the  sun  alters  the  eccentricity  of  the 
moon's  orbit,  and  differently  in  different  revolutions  of  the  moon, 
according  to  the  position  of  the  line  of  syzigies  with  respect  to 
the  line  of  apsides.  When  these  lines  are  coincident,  the  eccen- 
tricity is  increased.  For,  suppose  P  M  A  N  (Fig.  75)  to  be  the 
elliptic  orbit  of  the  moon  that  would  be  described  under  the 
influence  of  a  force  varying  inversely  as  the  square  of  the  dis- 
tance. In  going  from  the  apogee  to  the  perigee,  the  gravity 
will  increase  in  a  greater  ratio  than  that  of  the  inverse  square 


228  ASTRONOMY. 

of  the  distance  ;  the  true  orbit  will  therefore  fall  within  the 
ellipse,  and  the  perigean  distance  (E  P)  will  be  less  than  for  the 
ellipse.  Consequently,  the  eccentricity  will  increase  so  much 
the  more  as  the  major  axis  diminishes.  On  the  other  hand,  in 
going  from  the  perigee  to  the  apogee,  the  gravity  will  decrease 
in  a  greater  ratio  than  the  inverse  square  of  the  distance,  and 
the  moon  will  consequently  recede  farther  from  the  earth,  than 
if  it  were  not  disturbed  by  the  sun.  Therefore,  in  this  half  of 
the  orbit  the  eccentricity  will  also  be  increased.  When  the 
apsides  are  in  quadratures,  the  eccentricity  will  be  diminished  ; 
for,  the  gravity  will  then  vary  from  the  apogee  to  the  perigee, 
and  from  the  perigee  to  the  apogee,  in  a  less  ratio  than  that  of 
the  inverse  squares ;  and  therefore  the  results  will  be  contrary 
to  those  just  obtained.  The  eccentricity  will  have  its  maxi- 
mum value  when  the  apsides  are  in  syzigies,  and  its  minimum 
when  they  are  in  quadratures:  for,  in  every  other  position  of 
the  line  of  apsides  with  respect  to  the  line  of  syzigies,  the  radial 
force  in  the  apogee  and  perigee  will  be  less  than  in  these  posi- 
tions (equa.  137),  and  therefore  alter  less  the  proportional  gravity 
of  the  moon  in  the  apogee  and  perigee.  It  is  evident,  from  the 
gradual  decrease  of  the  radial  force,  as  we  recede  from  the  syzi- 
gies and  quadratures,  that  the  eccentricity  will  continually 
diminish  in  the  progress  of  the  apsides  from  the  syzigies  to  the 
quadratures,  and  that  it  will  continually  increase  from  the  quad- 
ratures to  the  syzigies. 

The  change  in  the  eccentricity  of  the  moon's  orbit,  thus  pro- 
duced, will  be  attended  with  a  corresponding  change  in  the  equa- 
tion of  the  centre,  and  thus  of  the  longitude.  And  this  change  is 
the  conspicuous  inequality  of  the  moon  known  by  the  name  of 
Evection,  (Art.  272). 

566.  The  radial  force  also  produces  a  motion  of  the  line  of  ap- 
sides. If  the  moon  was  only  acted  upon  by  the  attraction  of  the 
earth,  its  orbit  would  be  an  ellipse,  and  the  motion  from  one  apsis 
to  another,  or,  in  other  words,  from  one  point  where  the  orbit  cuts 
the  radius  vector  at  right  angles  to  the  other,  would  be  180^.  In 
point  of  fact,  however,  the  gravity  due  to  the  earth's  attraction  is 
constantly  either  diminished  or  increased  by  the  radial  disturbing 
force  of  the  sun,  and  therefore  its  true  orbit  must  continually  de- 
viate from  the  ellipse  that  would  be  described  under  the  sole 


EFFECTS    OF    THE    RADIAL    FORCE.  229 

action  of  the  earth's  attraction.  When  from  the  action  of  this 
force  there  is  a  diminution  of  the  moon's  gravity,  she  will  con- 
tinually recede  from  the  ellipse  in  question,  her  path  will  be 
less  bent,  and  she  must  therefore  move  through  a  greater  angu- 
lar distance,  before  the  central  force  will  have  deflected  her 
course  into  a  direction  at  right  ano-les  to  the  radius  vector. 
Accordingly  she  will  move  through  a  greater  angular  distance 
than  180°  in  going  from  one  apsis  to  another,  and  thus  the 
apsides  will  advance.  On  the  other  hand,  when  the  same 
force  increases  the  moon's  gravity,  her  path  will  fall  within  the 
ellipse,  its  curvature  will  be  increased,  and  therefore  it  will  be 
brought  to  intersect  the  radius  vector  at  right  angles,  at  a  less 
angular  distance.  In  this  case,  therefore,  the  apsides  will  move 
backwards.  Now,  we  have  shown,  (Art.  563),  that  the  radial 
disturbing  force  of  the  sun  alternately  diminishes  and  increases 
the  moon's  gravity  to  the  eartli.  It  follows,  therefore,  that  the 
motion  of  the  apsides  will  be  alternately  direct  and  retrograde  ; 
but  since,  as  has  been  shown,  (Art.  563),  the  diminution  sub- 
sists during  a  longer  part  of  the  moon's  revolution,  and  is  more- 
over greater  than  the  increase,  the  direct  motion  will  exceed  the 
retrograde,  and  therefore  in  an  entire  revolution  the  apsides  will 
advance. 

567.  The  observed  motion  of  the  apsides  of  the  moon's  orbit 
is  not,  however,  wholly  produced  by  the  radial  disturbing  force. 
It  is  in  part  due  to  the  action  of  the  tangential  force.  This 
force  alters  the  centrifugal  force  of  the  moon,  and  thus  changes 
its  gravity  towards  the  earth,  at  the  same  time  with  the  radial 
force. 

568.  The  elliptic  form  of  the  sun's  orbit  is  the  occasion  of  a 
change  in  the  radial  force,  from  which  results  a  perturbation  of 
longitude,  called  the  Annual  Equation  (Art.  272).  The  mean 
diminution  of  the  moon's  gravity,  arising  from  the  action  of  the 

sun,  or  the  mean   radial   force,   is   equal  to    (Art.   563). 

2  a^ 

Hence     this    diminution     is    inversely    proportional     to    the 

cube  of   the    sun's    distance    from  the   earth.     Therefore,    as 

the    sun    approaches    the    perigee   of  its    orbit,    its    distance 

from   the    earth    diminishing,    the    mean    diminution    of  the 

moon's  gravity  to  the  earth  will  increase,  and  consequently 


230  ASTRONOMY. 

the  moon's  distuiice  iVoiii  tlie  earth  will  become  greater,  and 
its  motion  slower,  than  it  otherwise  would  be.  The  con- 
trary will  take  place  while  the  sun  is  moving  from  the  perigee 
to  the  apogee. 

569.  The  disturbing  force  perpendicular  to  the  plane  of  the 
moon's  orbit,  produces  a  tendency  in  the  moon  to  quit  that  plane, 
from  which  there  results  a  change  in  the  position  of  the  line  of 
the  nodes,  and  a  change  in  the  inclination  of  the  plane  of  the  orbit 
to  that  of  the  ecliptic.  If  we  examine  the  general  expression  for 
this  force,  viz : 

f  3m y cos (p    .    „    .    , 

perpen. force  —  — -^ l  sm  IS  sm  I, 

a  2 

we  see  that  for  any  given  values  of  S  and  I,  it  will  be  zero  in  the 
quadratures,  and  have  its  greatest  value  in  the  syzigies  ;  and  that 
it  will  change  its  direction  in  the  quadratures,  lying,  in  the  nearer 
half  of  the  orbit,  on  the  same  of  its  plane  as  the  sun,  and  in  the 
more  remote  half,  on  the  opposite  side.  We  perceive  also  that  it 
will  be  zero  for  every  value  of  cp,  or  for  every  elongation  of  the 
moon,  when  the  angle  S  is  zero,  that  is,  when  the  sun  is  in  the 
plane  of  the  orbit,  and  will  attain  its  maximum,  for  any  given 
elongation,  when  the  line  of  direction  of  the  sun  is  perpendicular 
to  the  line  of  nodes.  It  will  also  be  the  less,  other  things  being 
the  same,  the  smaller  is  the  inclination  I. 

570.  Now,  let  N  M'  R  (Fig.  76)  represent  the  orbit  of  the  moon, 
and  S  the  sun,  supposed  stationary,  the  line  of  the  nodes  being  in 
quadratures  ;  and  let  L,  L'  be  the  points  of  the  orbit  90°  distant 
from  the  nodes.  The  direction  of  the  force,  in  the  various  points 
of  the  orbit,  is  indicated  by  the  arrows  drawn  in  the  figure.  When 
the  moon  is  at  any  point  M'  between  L  and  the  descending  node 
N'j  she  will  be  drawn  out  of  the  plane  in  which  she  is  moving  by 
the  disturbing  force  M'  K',  and  compelled  to  move  in  a  line  such  as 
M'  t'.  The  node  N'  will,  therefore,  retrograde  to  some  point  n'. 
When  she  is  at  any  point  M,  moving  from  the  ascending  node  N 
towards  L,  her  course  will  be  changed  to  the  line  M  t,  lying,  like 
the  line  M'  t',  below  the  orbit,  which  being  produced  backwards, 
meets  the  plane  of  the  ecliptic  in  some  point  n,  behind  N.  The 
nodes,  therefore,  retrograde  in  this  position  of  the  moon,  as  well 
as  in  the  former.  When  the  moon  is  in  the  half  N'  L'  N  of  the 
orbit,  lying  below  the  ecliptic,  the  absolute  direction  of  the  dis- 


EFFECTS    OF    THE    RADIAL    FORCE.  231 

turbing  force  will  be  reversed,  and  thus  its  tendency  will  be  the 
same  as  before,  namely,  to  draw  the  moon  towards  the  ecliptic. 
It  follows,  therefore,  that  throughout  this  half  of  the  orbit,  as  in 
the  other,  the  motion  of  the  nodes  will  be  retrograde.  Accord- 
ingly, when  the  nodes  are  in  quadratures,  or  90°  distant  from  the 
sun,  they  will  retrograde  during  every  part  of  the  revolution  of 
the  moon. 

571.  Suppose  the  sun  now  to  be  fixed  on  the  line  of  nodes,  or 
the  nodes  to  be  in  syzigies.  In  this  case  the  perpendicular  force 
will  be  zero,  (Art.  569),  and,  therefore,  there  will  be  no  disturb- 
ance of  the  plane  of  the  moon's  orbit. 

572.  Next,  let  the  situation  of  the  sun  be  intermediate  between 
the  two  just  considered,  as  represented  in  Figs.  76  and  77.  The 
effect  of  the  disturbing  force  will  be  the  same  as  in  the  first  situation 
from  the  quadrature  q  (Fig.  76)  to  the  node  N',  and  from  the  quad- 
rature q'  to  the  node  N.  But  throughout  the  arcs  N  q,  N'  q'  the 
direction  of  the  force,  and  therefore  the  effects,  will  be  reversed. 
The  node  will  then  retrograde,  as  before,  while  the  moon  moves 
over  the  arcs  q  N'  and  q'  N,  and  advance  while  she  is  in  the  arcs 
N  g-,  N'  q'.  But  as  the  force  is  greatest  over  the  arcs  q  N',  q'  N, 
wliich  contain  the  syzigies,  (Art.  ^69\  and  as  these  arcs  are  also 
longer  than  the  arcs  N  ^,  N'  q\  the  node  will,  on  the  whole,  retro- 
grade each  revolution.  The  velocity  of  retrogradation  will,  how- 
ever, be  less  than  when  the  nodes  are  in  quadratures,  and  pro- 
portionably  less,  as  the  distance  of  the  sun  from  this  position  is 
greater. 

In  the  position  represented  in  Fig.  77 ,  a  direct  motion  will 
take  place  over  the  arcs  q'  N'  and  q  N  ;  but  as  N  q\  and  N'  q,  the 
arcs  of  retrograde  motion,  are  of  greater  extent  than  q'  N',  and 
q  N,  and  moreover  contain  the  syzigies,  the  retrograde  motion  in 
each  revolution  must  exceed  the  direct,  as  before. 

If  we  suppose  the  sun  to  be  situated  on  the  other  side  of  the 
line  of  nodes,  the  effect  of  the  disturbing  force  will  obviously  be 
the  same  in  any  one  position  of  the  sun,  as  in  the  position  diame- 
trically opposite  to  it.  It  appears,  then,  that  the  line  of  the  nodes 
has  a  retrograde  motion  in  every  possible  position  of  the  sun. 

573.  We  have  thus  far  supposed  the  sun  to  remain  stationary  in 
the  various  positions  in  which  we  have  supposed  it,  during  the  revo- 
lution of  the  moon.     It  remains,  then,  to  consider  the  effect  of  the 


232  ASTRONOMY. 

sun's  motion  in  this  interval.  And,  first,  it  is  plain,  that,  as  the 
sun  advances  from  S  towards  N',  (Fig.  76),  the  arcs  N  </,  N'  <i  will 
increase,  and  the  arcs  ^N'  and  ^'N  diminish;  from  which  it  ap- 
pears, that,  during  the  advance  of  the  sun  from  the  point  90° 
behind  the  descending  node  to  this  node,  its  motion  in  the  course 
of  each  revolution  of  the  moon  will  cause  the  retrograde  motion  of 
the  node  to  be  slower  than  it  otherwise  would  be.  While  the  sun 
moves  from  the  ascending  node  to  the  90°  from  it,  the  effect  of  its 
motion  will  obviously  be  just  the  reverse  of  this.  During  its 
passage  from  the  descending  to  the  ascending  node,  the  effect  will 
be  the  same  in  either  quadrant,  as  in  that  diametrically  opposite. 

The  variation  in  the  intensity  of  the  perpendicular  force,  eon- 
spires  with  the  difference  of  situation  of  the  sun  and  its  motion 
during  a  revolution  of  the  moon,  in  diminishing  or  increasing,  as 
the  case  may  be,  the  velocity  of  retrogradation  of  the  nodes. 

574.  Let  us  now  treat  of  the  change  of  the  inclination  of  the 
orbit,  resulting  from  the  disturbing  action  of  the  sun.  And,  first, 
if  we  refer  to  Fig.  76,  we  shall  see  that  when  the  nodes  are  in 
quadrature,  the  inclination  will  diminish  while  the  moon  is 
moving  from  the  ascending  node  N  to  the  point  L  90°  distant  from 
it,  and  increase  while  she  is  moving  from  L  to  the  other  node  N'. 
In  the  other  half  of  the  orbit,  the  tendency  of  the  disturbing  force 
is  the  same,  (Art.  570) ;  and,  therefore,  while  the  moon  is  moving 
from  N'  to  L',  the  inclination  will  diminish,  and  while  she  is 
moving  from  L'  to  N,  it  will  increase.  The  diminutions  and  in- 
crements will  compensate  each  other,  and  the  original  inclination 
will  be  regained  at  the  close  of  the  revolution. 

575.  When  the  nodes  are  in  syzigies  there  will  be  no  change 
of  inclination  (Art.  569). 

576.  In  the  situations  of  the  sun,  represented  in  Figs.  76  and 
77,  the  inclination  will  decrease  from  9  to  L  and  from  ((  to  L', 
and  increase  from  L  to  </'  and  from  L'  to  9,  the  effects  being  the 
same  as  when  the  nodes  are  in  quadratures  over  the  arcs  q  L 
and  L  N'  in  Fig.  76,  and  N  L  and  L  9'  in  Fig.  77 ,  and  being 
reversed  over  the  arcs  N  q  and  N'  9'  in  Fig.  76,  and  q  N  and  <^  N' 
in  Fig.  77.  When  the  sun  has  the  position  represented  in  Fig. 
76,  the  arcs  of  increase  L  9'  and  L'  q  will  be  greater  than  the 
arcs  of  diminution  q  L  and  ((  L'.  The  disturbing  force  will  also 
be  greater  in  the  former  arcs  than  in  the  latter.     In  the  position 


EFFECTS  OF  THE  PERPENDICULAR  FORCE.        233 

supposed,  therefore,  there  will  be,  on  the  whole,  an  increase  of 
inclination  eveiy  revolution.  When  the  sun  is  in  the  position 
represented  in  Fig.  77,  the  arcs  of  diminution  q  L  and  q'  L'  will 
be  the  greater ;  and  the  force  in  them  will  also  be  the  greater. 
In  this  case,  therefore,  there  will  be  a  diminution  of  the  inclina- 
tion each  revolution  of  the  moon. 

When  the  sun  is  on  the  other  side  of  the  line  of  nodes,  the 
results  will  be  the  same  as  in  the  positions  diametrically  op- 
posite. 

577.  To  inquire  now  into  the  consequences  of  the  sun's  mo- 
tion during  the  revolution  of  the  moon.  As  the  sun  moves  from 
S,  towards  N'  (Fig.  76)  the  arcs  L  q',  L'  q,  over  which  there  is  an 
increase  of  the  inclination,  will  increase  ;  and  the  arcs  q  L,  q'  L', 
over  which  there  is  a  diminution,  will  diminish.  The  motion 
of  the  sun  will,  therefore,  in  approaching  the  descending  node, 
render  the  increase  of  the  inclination  each  revolution  of  the 
moon  greater  than  it  otherwise  would  be.  When  the  sun  is 
receding  from  the  ascending  node,  the  corresponding  arcs  will 
experience  corresponding  changes,  and  therefore  the  diminution 
will  now  be  less  than  if  the  sun  were  stationary. 

The  results  will  be  similar  for  the  opposite  quadrants  on  the 
other  side  of  the  line  of  nodes. 

578.  Since  the  inclination  diminishes  as  the  sun  recedes  from 
either  node,  and  increases  as  it  approaches  either  node,  it  will 
be  the  least  when  the  nodes  are  in  quadratures,  and  the  greatest 
when  they  are  in  syzigies. 

579.  The  perturbations  of  the  elliptic  motion  of  the  moon, 
comprising  inequalities  of  orbit  longitude,  and  variations  in  the 
form  and  position  of  the  orbit,  which  have  now  been  under  con- 
sideration, depend  upon  the  configurations  of  the  sun  and  moon, 
with  respect  to  each  other,  the  perigee  of  each  orbit,  and  the 
node  of  the  moon's  orbit.  Their  effects  will  disappear  when 
the  configurations  upon  which  they  depend  become  the  same. 
They  are  therefore  periodical. 

580.  The  perturbations  of  the  motions  of  a  planet,  produced 
by  the  action  of  another  planet,  are  precisely  analogous  to  the 
perturbations  of  the  motions  of  the  moon,  produced  by  the  ac- 
tion of  the  sun.     The  disturbing  forces  are  obviously  of  the 

30 


234  ASTRONOMY. 

same  kind,  and  they  are  subject  to  variations  from  precisely 
similar  causes.  But,  owing  to  the  smallness  of  the  masses  of 
the  planets  and  their  great  distances,  their  disturbing  forces  are 
much  more  minute  than  the  disturbing  force  of  the  sun.  From 
this  cause,  together  with  the  slow  relative  motion  of  the  disturb- 
ing  and  disturbed  body,  the  motion  of  the  apsides  and  nodes, 
and  the  accompanying  variations  of  eccentricity  and  inclina- 
tion, are  very  much  more  gradual  in  the  case  of  the  planets 
than  in  the  case  of  the  moon.  Their  periods  comprise  many 
thousands  of  years,  and  on  this  account  they  are  called  Sectdar 
Motions  or  Variations.  In  consequence  of  the  greater  feeble- 
ness of  the  disturbing  forces,  the  periodical  inequalities  are  also 
much  less  in  amount.  Moreover,  as  the  motion  of  a  planet  is 
much  slower  than  that  of  the  moon,  and  as  the  variat  ons  of  its 
orbit  are  more  gradual  than  those  of  the  lunar  orbit,  the  com- 
pensations produced  by  a  change  of  configurations  are  much 
more  slowly  effected,  and  thus  the  periods  of  the  inequalities  are 
much  longer. 

581.  The  motions  of  the  moon  would  be  subject  to  no  secu- 
lar variations,  if  the  apparent  orbit  of  the  sun  were  unchange- 
able ;  but  the  secular  variation  of  the  eccentricity  of  the  sun's 
orbit,  which  answers  to  an  equal  variation  of  the  eccentricity  of 
the  earth's  orbit,  that  is  produced  by  the  action  of  the  planets, 
gives  rise  to  a  secular  inequality  in  the  motion  of  the  moon, 
called  the  Acceleration  of  the  Moon.  This  inequality  was  dis- 
covered from  observation.  Its  physical  cause  was  first  made 
known  by  Laplace. 


MASSES    OF    THE    SUN,    MOON,    AND    PLANETS.  235 


CHAPTER    XXIII. 

OP  THE  RELATIVE  MASSES  AND  DENSITIES  OF  THE  SUN, 
MOON,  AND  PLANETS  ;  AND  OF  THE  RELATIVE  INTENSITY 
OP    THE    GRAVITY    AT    THEIR    SURFACE. 

582.  The  perturbations  which  a  planet  produces  in  the  mo- 
tions of  the  other  planets,  depend  for  their  amount  chiefly  upon 
the  ratio  of  the  mass  of  the  planet  to  the  mass  of  the  sun,  and 
the  ratio  of  the  distance  of  the  planet  from  the  sun  to  the  dis- 
tance of  the  planet  disturbed  from  the  same  body.  Now,  the 
ratio  of  the  distances  is  known  by  the  methods  of  Plane  Astro- 
nomy ;  consequently,  the  observed  amount  of  the  perturbations 
ought  to  make  known  the  ratio  of  the  masses,  the  only  unknown 
element  upon  which  it  depends. 

This  is  one  method  of  determining  the  masses  of  the  planets. 
The  masses  of  those  planets  which  have  satellites  maybe  found 
by  another  and  simpler  method.  The  theory  of  gravitation  has 
furnished  the  following  equation,  from  which  they  may  easily 
be  derived  : 

M  +  m   ^   d^  P2 
1 4-M        D=»p2' 

in  which  1  denotes  the  mass  of  the  sun,  M  the  mass  of  the  planet, 
m  the  mass  of  one  of  its  satellites,  D  the  mean  distance  of  the 
planet  from  the  sun,  d  the  mean  distance  of  the  satellite  from  the 
planet,  and  P  and  p  the  periodic  times  of  the  planet  and  satellites 
respectively.  As  the  mass  of  the  satellite  is  small  compared  with 
that  of  the  planet,  and  the  mass  of  the  planet  is  small  compared 
with  that  of  the  sun,  they  may  be  neglected  in  the  above  equation 
with  but  little  error ;  and  thus  we  have, 

M  =  — (very  nearly). 

]33p2   V        ^  ■'^ 

583.  The  second  column  of  Table  IV  exhibits  the  relative 
masses  of  the  sun,  moon,  and  planets,  according  to  the  most 
received  determinations,  that  of  the  earth  being  denoted  by  1. 

684.  The  quantities  of  matter  of  the  sun,  moon,  and  planets,  as 


236  ASTRONOMY. 

well  as  their  bulks,  being  known,  their  densities  may  be  easily 
computed ;  for,  the  densities  of  bodies  are  proportional  to  their 
quantities  of  matter  divided  by  their  bulks.  The  third  column 
of  Table  IV  contains  the  densities  of  the  sun,  moon,  and  planets, 
that  of  the  earth  being  denoted  by  1.  It  will  be  seen  on  inspect- 
ing it,  that,  for  the  most  part,  the  densities  of  the  planets  decrease 
as  we  recede  from  the  sun. 

585.  The  relative  intensity  of  the  gravity  at  the  surface  of  the 
sun,  moon,  and  planets,  may  also  readily  be  found,  when  the 
masses  and  bulks  of  these  bodies  are  known.  For,  supposing 
them  to  be  spherical,  and  not  to  rotate  on  their  axes,  the  gravity 
at  their  surface  will  be  directly  as  their  masses,  and  inversely  as 
the  squares  of  their  radii,  or,  in  other  words,  proportional  to  their 
masses  divided  by  the  squares  of  their  radii.  The  centrifugal 
force  at  the  surface  of  a  planet,  generated  by  its  rotation  on  its 
axis,  diminishes  the  gravity  due  to  the  attraction  of  the  matter  of 
the  planet.  The  diminution  thus  produced  on  any  of  the  planets 
is  not,  however,  very  considerable.  The  method  of  determining 
the  centrifugal  force  at  the  surface  of  a  body  in  rotation,  is  given 
in  treatises  on  Mechanics.  (See  Courtenay's  Mechanics,  pages 
250  and  251). 

The  fourth  column  of  Table  IV  exhibits  the  relative  intensity 
of  the  gravity  at  the  surface  of  the  sun,  moon,  and  planets,  that 
at  the  surface  of  the  earth  being  denoted  by  1. 


CHAPTER    XXIV. 

OF  THE    FIGURE  AND    ROTATION   OF   THE    EARTH;    AND   OF   THE 
PRECESSION    OF    THE    EQUINOXES    AND    NUTATION. 

586.  We  have  already  seen  (Art.  145)  that  measurements  made 
upon  the  earth's  surface,  establish  that  the  figure  of  the  earth  is 
that  of  an  oblate  spheroid,  and  that  the  oblateness  at  the  poles  is 
about  j^j. 


EXPLANATION   OF   SPHEROIDAL   FORM   OF  THE    EARTH.       237 

587.  From  the  amount  and  law  of  the  variation  of  the  force 
of  gravity  upon  the  earth's  surface,  ascertained  by  observations 
upon  the  length  of  the  seconds'  pendulum,  it  is  proved  that  the 
matter  of  the  earth  is  not  homogeneous,  but  denser  towards  the 
centre,  and  that  it  is  arranged  in  concentric  strata  of  nearly  an 
elliptical  form  and  uniform  density. 

The  fact  of  the  greater  density  of  the  earth  towards  its  centre, 
has  also  been  established  by  observations  upon  the  deviation  of  a 
plumb  line  from  the  vertical,  produced  by  the  attraction  of  a 
mountain.  Such  observations  were  made  for  the  purpose  of 
determining  the  mean  density  of  the  earth  by  Dr.  Maskelyne,  on 
the  sides  of  the  mountain  Schehallien  in  Scotland.  The  ob- 
served deviation  of  the  plumb  line  made  known  the  ratio  of  the 
attraction  of  the  mountain  to  that  of  the  whole  earth,  and  thus  the 
relative  quantities  of  matter  in  the  mountain  and  earth.  These 
being  ascertained,  and  the  figure  and  bulk  of  the  mountain  hav- 
ing been  determined  by  a  survey,  the  relative  density  of  the  earth 
and  mountain  became  known  by  the  prmciple  mentioned  in 
Art.  584,  and  thence  the  actual  density  of  the  earth,  the  density 
of  the  mountain  having  been  found  by  experiment.  The  result 
was,  that  the  mean  density  of  the  earth  is  4.95,  the  density  of 
water  being  1. 

588.  The  spheroidal  form  of  the  surface  of  the  earth  and  of  its 
internal  strata  is  easily  accounted  for,  if  we  suppose  the  earth  to 
have  been  originally  in  a  fluid  state.  The  tendency  of  the  mutual 
attraction  of  its  particles  would  be  to  give  it  a  spherical  form  ;  but, 
by  virtue  of  its  rotation,  all  its  particles,  except  those  lying  imme- 
diately on  the  axis,  would  be  animated  by  a  centrifugal  force  increas- 
ing with  their  distance  from  the  axis.  If,  therefore,  we  conceive  of 
two  columns  of  fluid  extending  to  the  earth's  centre,  one  from  near 
the  equator,  and  the  other  from  near  either  pole,  the  weight  of  the 
former  would  by  reason  of  the  centrifugal  force  be  less  than  that 
of  the  latter.  In  order,  then,  that  they  may  sustain  each  other  in 
equilibrio,  that  near  the  equator  must  increase  in  length,  and  that 
near  the  pole  diminish.  As  this  would  be  true  at  the  same  time  for 
every  pair  of  columns  situated  as  we  have  supposed,  the  surface 
of  the  whole  body  of  fluid  about  the  poles  must  fall,  and  that  of  the 
fluid  about  the  equator  rise.  In  this  manner  the  earth  would  be- 
come flattened  at  the  poles  and  protuberant  at  the  equator. 


238  ASTRONOMY. 

589.  Upon  a  strict  investigation  it  appears  that  a  homogeneous 
fluid  of  the  same  mean  density  with  the  earth,  and  rotating  on  its 
axis  at  the  same  rate  that  the  earth  does,  would  be  in  equihbrium, 
if  it  had  the  figure  of  an  oblate  spheroid,  of  which  the  axis  was 
to  the  equatorial  diameter  as  229  to  230,  or  of  which  the  oblate- 
ness  was  ^i^.  If  the  fluid  mass  supposed  to  rotate  on  its  axis  be 
not  homogeneous,  but  be  composed  of  strata  that  increase  in  den- 
sity from  the  surface  to  the  centre,  the  solid  of  equilibrium  will 
still  be  an  elliptic  spheroid,  but  the  oblateness  will  be  less  than 
when  the  fluid  is  homogeneous. 

590.  The  time  of  the  earth's  rotation,  as  well  as  the  position  of 
its  axis,  would  change  if  any  variation  should  take  place  in  the 
distribution  of  the  matter  of  the  earth,  or  in  case  of  the  impact  of 
a  foreign  body. 

If  any  portion  of  matter  be,  from  any  cause,  made  to  approach 
the  axis,  its  velocity  will  be  diminished,  and  the  velocity  lost  being 
imparted  to  the  mass,  will  tend  to  accelerate  the  rotation.  If  any 
portion  of  matter  be  made  to  recede  from  the  axis,  the  opposite 
effect  will  be  produced,  or  the  rotation  Avill  be  retarded.  In  point 
of  fact,  the  changes  that  take  place  in  the  position  of  the  matter 
of  the  earth,  whether  from  the  washing  of  rains  upon  the  sides  of 
mountains,  or  evaporation,  or  any  other  known  cause,  are  not 
sufficient  ever  to  produce  any  sensible  alteration  in  the  circum- 
stances of  the  earth's  rotation  on  its  axis. 

591.  It  is  ascertained  from  direct  observation,  that  there  has  in 
reality  been  no  perceptible  change  in  the  period  of  the  earth's 
rotation  since  the  time  of  Hipparchus,  120  years  before  the 
beginning  of  the  present  era.  We  may  therefore  conclude, 
a  'posteriori,  that  there  has  been  no  material  change  in  the  form 
and  dimensions  of  the  earth  in  this  interval. 

592.  Were  the  axis  of  the  earth  to  experience  any  change  of 
position  with  respect  to  the  matter  of  the  earth,  the  latitudes  of 
places  would  be  altered.  A  motion  of  200  feet  might  increase  or 
diminish  the  latitude  of  a  place  to  the  amount  of  2",  an  angle 
which  can  be  measured  by  modern  instruments.  Now,  in  point 
of  fact,  the  latitudes  of  places  have  not  sensibly  varied  since  their 
first  determination  with  accurate  instruments ;  therefore,  in  this 
interval  the  axis  of  the  earth  cannot  have  changed.  Indeed,  since 
the  earth's  surface  and  its  internal  strata  are  arranged  symmetri- 


PHYSICAL  THEORY  OF  PRECESSION  AND    NUTATION.       239 

cally  with  respect  to  the  present  axis  of  rotation,  it  is  to  be  in- 
ferred that  this  axis  is  the  same  as  that  which  obtained  at  the 
epoch  when  the  matter  of  the  earth  changed  from  a  fluid  to  a 
sohd  state. 

593.  The  motions  of  the  earth's  axis,  along  with  the  whole 
body  of  the  earth,  which  give  rise  to  the  Precession  of  the  Equi- 
noxes and  Nutation,  are  consequences  of  the  spheroidal  form  of 
the  earth,  inasmuch  as  they  are  produced  by  the  actions  of  the 
sun  and  moon  upon  that  portion  of  the  matter  of  the  earth  which 
lies  on  the  outside  of  a  sphere  conceived  to  be  described  about 
the  earth's  axis.  The  physical  theory  of  the  phenomena  in 
question  is  analogous  to  that  of  the  retrogradation  of  the  moon's 
nodes.  The  sun  produces  a  retrograde  movement  of  the  points 
in  which  the  circle  described  by  each  particle  of  the  protuberant 
mass  cuts  the  plane  of  the  ecliptic,  as  it  does  of  the  moon's 
nodes ;  the  effect  produced  is,  however,  exceedingly  small,  by 
reason  of  the  inertia  of  the  interior  spherical  mass  connected 
with  the  extenal  mass  upon  which  the  action  takes  place.  The 
moon,  in  like  manner,  occasions  a  retrograde  movement  of  the 
nodes  of  the  same  particles  on  the  plane  of  its  orbit.  The  ac- 
tions of  the  sun  and  moon  will  not  be  the  same  each  revolution 
of  a  particle.  That  of  the  sun  will  vary  during  the  year  with 
the  angular  distance  of  the  sun  from  the  node  (Art.  569) ;  and 
that  of  the  moon  will  vary  during  each  month  with  the  distance 
of  the  moon  from  the  node,  and  also  during  a  revolution  of  the 
nodes  of  the  moon's  orbit  by  reason  of  the  change  in  the  inclina- 
tion of  the  orbit  to  the  equator.  The  mean  effect  of  both  bodies 
is  the  precession  ;  the  inequality  resulting  from  the  change  in 
the  sun's  action  during  the  year  is  the  solar  nutation  ;  and  the 
inequality  consequent  upon  the  retrogradation  of  the  moon's 
nodes  is  the  lunar  nutation ;  or  the  chief  part  of  it :  the  change 
in  the  position  of  the  equinox  occasioned  by  the  moon's  revolu- 
tion, never  exceeds  |^  of  a  second  of  an  arc  ;  and  the  change  of 
the  obliquity  of  the  ecliptic  from  this  cause  is  still  less. 


240  ASTRONOMY. 


CHAPTER    XXV 


OF    THE    TIDES. 


594.  The  alternate  rise  and  fall  of  the  surface  of  the  ocean 
twice  in  the  course  of  a  lunar  day,  or  about  25  hours,  is  the 
phenomenon  known  by  the  name  of  the  Tides.  The  rise  of 
the  water  is  called  the  Flood  Tide,  and  the  fall  the  Ehh  Tide. 

595.  The  interval  between  one  high  water  and  the  next,  is, 
at  a  mean,  half  a  mean  lunar  day,  or  12h.  25m.  14s.  Low  wa- 
ter has  place  nearly,  but  not  exactly,  at  the  middle  of  this  inter- 
val ;  the  tide,  in  general,  employing  nine  or  ten  minutes  more 
in  ebbing  than  in  flowing.  As  the  interval  between  one  period 
of  high  water  and  the  second  following  one  is  a  lunar  day,  or 
Id.  Oh.  50m.  28s.,  the  retardation  in  the  time  of  high  water 
from  one  day  to  another  is  50m.  28s.,  in  its  mean  state. 

596.  The  time  of  high  water  is  mainly  dependent  upon  the 
position  of  the  moon,  being  always  at  any  given  place  about  the 
same  length  of  time  after  the  moon's  passage  over  the  superior 
or  inferior  meridian.  As  to  the  length  of  the  interval  between 
the  two  periods,  at  different  places,  in  the  open  sea  it  is  only 
from  two  to  three  hours  ;  but  on  the  shores  of  continents,  and  in 
rivers,  where  the  water  meets  with  obstructions,  it  is  very  differ- 
ent at  different  places,  and,  in  some  instances,  is  of  such  length 
that  the  time  of  high  water  seems  to  precede  the  moon's 
passage. 

597.  The  height  of  the  tide  at  high  water  is  not  alwa3''s  the 
same,  but  varies  from  day  to  day ;  and  these  variations  have  an 
evident  relation  to  the  phases  of  the  moon.  It  is  greatest  at  the 
syzigies  ;  after  which  it  diminishes  and  becomes  the  least  at  the 
quadratures. 

598.  The  tides,  about  the  time  of  the  syzioies,  are  called  the 
Spring  Tides  ;  and  those  about  the  time  of  quadratures  are 
called  the  Neap  Tides. 

The  highest  of  the  spring  tides  is  not  that  which  has  place 
nearest  the  syzigy,  but  is  in  general  the  third,  and  in  some  in- 


PHENOMENA    OF    THE    TIDES,  241 

Stances,  the  fourth  following  tide.     In  like  manner  the  lowest  of 
the  neap  tides  is  the  third  or  fourth  tide  after  the  quadrature. 

The  spring  tides  are,  in  general,  about  twice  the  heisfht  of  the 
neap  tides.  At  Brest,  in  France,  the  former  rise  to  the  height  of 
19.3  feet,  and  the  latter  only  to  9.2  feet.  In  the  Pacific  Ocean 
the  highest  of  the  tides  of  the  syzigies  is  5  feet,  and  the  lowest  of 
the  tides  of  the  quadratures  is  between  2  and  2.5  feet. 

599.  The  tides  are  also  affected  by  the  declinations  of  the 
sun  and  moon  ;  for  they  diminish  the  tides  of  the  syzigies, 
which  occur  at  the  equinoxes  ;  and  augment  the  tides  of  the 
quadratures,  which  occur  at  the  solstices.  Also,  when  the  moon 
or  the  sua  are  out  of  the  equator,  the  evening  and  morning 
tides  differ  somewhat  in  height.  At  Brest,  in  the  syzisfies  of  the 
summer  solstice,  the  tides  of  the  morning  of  the  first  and  second 
day  after  the  syzigy  are  smaller  than  those  of  the  eveninsf  by 
6.6  inches.  They  are  greater  by  the  same  quantity,  in  the  syzi- 
gies of  the  summer  solstice. 

600.  The  distance  of  the  moon  from  the  earth  has  also  a  sensi- 
ble influence  upon  the  tides.  In  general,  they  increase  and 
diminish  as  the  distance  increases  or  diminishes,  but  in  a  more 
rapid  ratio. 

601.  The  daily  retardation  of  the  time  of  high  water  varies 
with  the  phases  of  the  moon.  It  is  at  its  minimum  towards  the 
syzigies,  when  the  tides  are  at  their  maximum  ;  and  it  is  then 
about  40m.  But,  towards  the  quadratures,  when  the  tides  are 
at  their  minimum,  the  retardation  is  the  greatest  possible  ;  and 
amounts  to  about  Ih.  15m. 

The  variation  in  the  distance  of  the  sun  and  moou  from  the 
earth,  (and  particularly  the  moon),  has  an  influence  also  on  this 
retardation. 

The  daily  retardation  of  the  tides  varies  likewise  with  the  de- 
clination of  the  sun  and  moon. 

602.  The  facts  which  have  been  enumerated  clearly  indicate 
that  the  tides  are  produced  by  the  actions  of  the  sun  and  moon 
upon  the  waters  of  the  ocean  ;  but  in  a  greater  degree  by  the 
action  of  the  moon.  To  explain  them,  let  us  suppose  at  first 
that  the  whole  surface  of  the  earth  is  covered  with  water.  It  has 
been  shown  (Art.  563)  that  the  sun's  action  increases  or  dimin- 

31 


242  ASTRONOMY. 

ishes  the  moon's  gravity  to  the  earth,  according  to  her  position 
with  respect  to  the  line  of  syzigies  or  quadratures,  or,  in  other 
words,  according  to  her  elongation  from  the  sun.  In  a  similar 
manner  will  the  moon's  action  increase  or  diminish  the  gravity  of 
a  particle  of  matter  at  the  earth's  surface,  according  to  its  elonga- 
tion from  the  moon,  as  seen  from  the  earth's  centre ;  for,  any 
particle  of  matter  upon  the  earth's  surface  is  attracted  towards  the 
earth's  centre,  just  as  if  the  whole  mass  of  the  earth  were  concen- 
trated at  its  centre ;  and  the  whole  earth  is  attracted  by  the  moon, 
just  as  it  would  be  if  its  matter  were  concentrated  at  its  centre. 
If  we  conceive  a  plane  to  pass  through  the  centre  of  the  earth,  at 
right  angles  to  the  line  joining  the  centres  of  the  earth  and  moon  ; 
within  about  35°  of  this  plane  on  each  side,  the  gravity  at  the  sur- 
face will  be  increased  (Art.  563) ;  and  at  the  remaining  parts,  that 
is,  for  about  55°  around  the  points  in  which  the  line  of  the  cen- 
tres intersects  the  surface,  the  gravity  will  be  diminished. 

The  equilibrum  of  the  waters  upon  the  surface  of  the  earth 
will  in  consequence  be  disturbed.  Around  the  points  just  men- 
tioned, where  the  gravity  is  diminished,  they  will  rise,  and  in  the 
middle  parts  between  these  points,  where  the  gravity  is  increased, 
they  will  fall.  In  consequence  of  the  earth's  diurnal  rotation,  the 
parts  of  the  surface,  at  which  the  rise  and  fall  of  the  water  will 
take  place,  will  be  continually  changing.  Were  the  entire  rise 
and  fall  produced  instantaneously,  the  points  of  highest  water 
would  constantly  be  the  precise  points  in  which  the  line  of  the 
centres  of  the  moon  and  earth  intersects  the  surface,  and  it  would 
always  be  high  water  on  the  meridian  passing  through  these 
points,  both  in  the  hemisphere  where  the  moon  is,  and  in  the 
opposite  one.  On  the  west  side  of  this  meridian,  the  tide  would 
be  flowing ;  on  the  east  side  of  it,  it  would  be  ebbing  ;  and  on  the 
meridian  at  right  angles  to  the  same,  it  would  be  low  water.  But 
it  is  plain  that  the  effects  of  the  moon's  action  will  not  be  instan- 
taneously produced,  and  therefore  that  the  points  of  highest  water 
will  fall  behind  the  moon.  It  appears  from  observation,  that  in 
the  open  sea  the  meridian  of  high  water  is  about  30°  to  the  east 
of  the  moon. 

The  great  tide  wave  thus  raised  by  the  moon,  and  which  fol- 
lows it  in  its  diurnal  motion,  will  be  a  mere  undulation,  or  alter- 
nate rise  and  fall  of  the  water,  without  any  progressive  motion. 


PHYSICAL  THEORY   OF  THE  TIDES.  243 

if,  as  we  have  supposed,  it  is  no  where  obstructed  by  shallows, 
islands,  or  the  shores  of  continents. 

603.  It  is  evident  that  the  sun  will  produce  precisely  similar 
effects  with  the  moon,  and  will  raise  a  tide  wave  similar  to  the 
lunar  tide  wave,  which  will  follow  it  in  its  diurnal  motion. 

604.  To  show  that  the  effects  of  the  sun  are  less  in  degree  than 
those  of  the  moon,  let  us  take  the  general  expression  for  the 
change  of  the  moon's  gravity  arising  from  the  action  of  the  sun, 
namely, 

—  X  y  (1  —  3  cos2  (p)  .  .  .  {a), 

in  which  m  denotes  the  mass  of  the  sun,  a  its  distance  (the  mean 
distance  of  the  moon  being  taken  as  1),  y  the  distance  of  the  moon 
in  its  given  position,  and  9  its  elongation  from  the  sun  as  seen 
from  the  earth's  centre.  This  formula  will  serve  to  express  the 
change  in  the  gravity  of  a  particle  of  matter  upon  the  earth's 
surface,  produced  by  the  sun's  action,  if  we  take  w  =  the  mass  of 
the  sun,  as  before,  a  =  its  distance  expressed  in  terms  of  the  radius 
of  the  earth  as  unity,  y  =  the  distance  of  the  particle  from  the  cen- 
tre of  the  earth,  and  cp  =  its  elongation  from  the  sun  as  seen  from 
the  earth's  centre.  If  we  designate  the  corresponding  quantities 
for  the  moon  by  m,',  a\  y,  (p,  we  shall  have  for  the  change  of  the 
gravity  of  a  particle,  produced  by  the  moon's  action, 

~Xy(l-3cos=^cp).  .  .(&). 

For  particles  at  equal  elongations  from  the  sun  and  moon,  we 
shall  have  cp  the  same  in  expressions  (a),  and  (6),  and  y  may 
be  regarded  as  the  same  without  material  error.  For  such  parti- 
cles, then,  the  alterations  of  the  gravity,  produced  by  the  sun  and 
moon,  will  bear  the  same  ratio  to  each  other  as  the  quantities 

and Now,  if  we  give  to  m,,  m',  a,  a',  their  values,  we 

shall  find  that  the  latter  quantity  is  nearly  three  times  greater  than 
the  former.  Accordingly,  the  effect  of  the  moon's  action,  at  cor- 
responding elongations  of  the  particles,  and  therefore  generally,  is 
nearly  three  times  greater  than  that  of  the  sun. 

605.  The  actual  tide  will  be  produced  by  the  joint  action  of  the 
sun  and  moon,  or  it  may  be  regarded  as  the  result  of  the  combi- 
nation of  the  limeir  and  solar  tide  waves. 


244  ASTRONOMY. 

At  the  time  of  the  syzigies,  the  action  of  the  sun  and  moon  will 
be  combined  in  producing  the  tides,  both  bodies  tending  to  produce 
high  as  well  as  low  water  at  the  same  places.  But  at  the  quadra- 
tures they  will  be  m  opposition  to  each  other,  the  one  tending  to 
raise  the  surface  of  the  water  where  the  other  tends  to  depress  it, 
and  vice  versa.  The  tides  should,  therefore,  be  much  higher  at 
the  syzigies  than  at  the  quadratures. 

Between  the  syzigies  and  the  quadratures  the  two  bodies  will 
neither  directly  conspire  with  each  other,  nor  directly  oppose  each 
other,  and  tides  of  intermediate  height  will  have  place.  The 
points  of  highest  water  will  also,  in  the  configuration  supposed, 
neither  be  the  vertices  of  the  lunar  nor  of  the  solar  tide  wave,  but 
certain  points  between  them.  This  circumstance  will  occasion  a 
variation  in  the  length  of  the  interval  between  the  time  of  the 
moon's  passage  and  the  time  of  high  water. 

606.  The  effect  of  the  moon's  action  being  to  that  of  the  sun's 
nearly  as  3  to  1,  (Art.  604),  the  spring  tides  will  be  to  the  neap 
tides  nearly  as  2  to  1.  For,  let  x  =  the  effect  of  the  moon,  and 
y  =  the  effect  of  the  sun  :  then  the  ratio  of  x  +  y  to  a:  —  y  will 
be  the  ratio  of  the  heights  of  the  spring  and  neap  tides.     Now, 

x  =  3y,  and  thus  ^L±^  ^  ll±l  =  2. 

X  —  y    3y  —  y 

This  result  is  conformable  to  observation. 

607.  The  height  of  the  tid?.,  as  well  as  the  interval  between 
the  time  of  high  water  and  that  of  the  moon's  meridian  passage, 
will  vary  not  only  with  the  elongation  of  the  moon  from  the  sun, 
but  also  with  the  distance  and  declination  of  the  moon  and  sun. 
For,  expressions  (a)  and  (6)  show  that  the  intensities  of  the 
moon's  and  sun's  actions  vary  inversely  as  the  cube  of  their  dis- 
tance ;  and  the  changes  of  the  declinations  of  the  two  bodies 
must  be  attended  with  a  change  both  in  the  absolute  and  relative 
situation  of  the  vertices  of  the  lunar  and  solar  tide  waves. 

608.  The  laws  of  the  tides,  which  would  obtain  on  the  hypothe- 
sis of  the  earth  being  covered  entirely  with  water,  are  found  to 
correspond  only  partially  with  those  of  the  actual  tides.  The 
continents  have  a  material  influence  upon  the  formation  and 
propagation  of  the  tide  wave. 

609.  Professor  Whewell  infers,  from  a  careful  discussion  of  a 
great  number  of  observations  upon  the  tides,  that  the  tide  of  the 


TIDES    OF    CHANNELS    AND    RIVERS.  245 

Atlantic  Ocean  is,  for  the  most  part,  prodnced  by  a  derivative  tide 
wave,  sent  off  from  the  great  wave  which  in  the  Sonthern  Ocean 
follows  the  moon  in  its  dinrnal  motion  around  the  earth.  This 
wave  advances  more  rapidly  in  the  open  sea  than  along  the  coasts, 
where  it  meets  with  obstructions. 

Where  portions  of  the  tide  wave,  extending  from  one  point  of 
the  coast  to  another,  become  detached,  and  advance  into  a  nar- 
row space,  particularly  high  tides  will  occur.  In  this  way  (as 
it  is  supposed)  it  happens  that  the  tide  rises  at  certain  places  in 
the  bay  of  Fundy,  to  the  height  of  60  or  70  feet. 

610.  In  channels  peculiar  tides  occur  in  consequence  of  the 
meeting  of  the  waves  which  enter  the  channels  at  their  two  ex- 
tremities. Where  the  two  waves  meet  in  the  same  state,  unusu- 
ally high  tides  occur.  This  is  observed  to  be  the  case  at  some 
points  in  the  Irish  Channel.  In  the  port  of  Batsha,  in  Tonquin, 
the  tides  arrive  by  two  channels,  of  such  lengths  that  the  two 
waves  meet  in  opposite  states,  or  that  the  flood  tide  arrives  by 
one  channel  just  as  the  ebb  tide  begins  to  leave  by  the  other, 
and  the  consequence  is,  that  there  is  neither  high  nor  low 
water. 

This  is  the  case  when  the  moon  is  in  the  equator.  When 
she  has  a  northern  or  southern  declination,  there  is  a  small  rise 
and  fall  of  the  water  once  in  a  lunar  day,  owing  to  the  inequality 
of  the  morning  and  evenino;  tides  of  the  open  sea. 

611.  Lakes  and  inland  seas  have  no  perceptible  tides,  for  the 
reason  that  their  extent  is  not  sufficient  to  admit  of  any  sensible 
inequality  of  gravity,  as  the  result  of  the  action  of  the  moon. 

612.  The  tides  experienced  in  rivers  and  seas  caminunicating 
with  the  ocean,  are  not  produced  by  the  direct  actions  of  the 
sun  and  moon,  but  are  waves  propagated  from  the  great  wave 
of  the  open  sea. 

In  rivers  of  considerable  length,  the  ascending  tides  are  en- 
countered by  those  which  are  reluming,  so  that  a  great  variety 
of  tides  occur  along  their  shores. 

613.  The  mean  interval  between  noon  and  the  time  of  high 
water  at  any  port,  on  the  day  of  new  or  full  moon,  is  called  the 
Establishment  of  that  port.  It  will  be,  approximately,  the  in- 
terval between  the  time  of  the  meridian  passage  of  the  moon 
and  the  time  of  high  water  on  any  day  of  the  month.     To  ob- 


246  ASTRONOMY. 

tain  this  interval  for  a  given  day  more  nearly,  it  is  necessary  to 
correct  the  establishment  for  the  effects  of  the  change  of  the  dis- 
tance and  declination  of  the  sun  and  moon,  and  of  the  change 
in  the  elongration  of  the  moon  from  the  sun.  When  it  has  been 
determined,  by  adding  it  to  the  time  of  the  meridian  passage  of 
the  moon,  we  have  the  time  of  the  next  high  water. 


PART   IV. 

ASTRONOMICAL    PROBLEMS. 

EXPLANATIONS  OF  THE  TABLES. 

The  Tables  which  form  a  part  of  this  work,  and  which  are 
employed  in  the  resolution  of  the  following  Problems,  consist  of 
Tables  of  the  Sun,  Tables  of  the  Moon,  Tables  of  the  Mean 
Places  of  some  of  the  Fixed  Stars.  Tables  of  Corrections  for 
Refraction,  Aberration  and  Nutation,  and  Auxiliary  Tables. 

The  Tables  of  the  Sun,  which  are  from  XVII  to  XXXIV  in- 
clusive, are,  for  the  most  part,  abridged  from  Delambre's  Solar 
Tables.  The  mean  longitudes  of  the  sun  and  of  his  perigee  for 
the  beginning  of  each  year,  found  in  Table  XVIII,  have  been 
computed  from  the  formulae  of  Prof  Bessel,  given  in  the  Nauti- 
cal Almanac  of  1837.  The  Table  of  the  Equation  of  Time 
was  reduced  from  the  table  in  the  Connaissance  des  Tems  of 
1810,  which  is  more  accurate  than  Delambre's  Table,  this  being 
in  some  instances  liable  to  an  error  of  2  seconds.  The  Table 
of  Nutation  (Table  XXVII)  was  extracted  from  Francoeur's 
Practical  Astronomy.  The  maximum  of  nutation  of  obliquity 
is  taken  at  9".25.  The  Tables  of  the  Sun  will  give  the  sun's 
longitude  from  the  mean  equinox  within  a  fraction  of  a  second 
of  the  result  obtained  immediately  from  Delambre's  Tables,  as 
corrected  by  Bessel.  The  Tables  of  the  Moon,  which  are  from 
XXXV  to  LXXXV  inclusive,  are  abridged  and  computed  from 
Burckhardt's  Tables  of  the  Moon.  To  facilitate  the  determina- 
tion of  the  hourly  motions  in  longitude  and  latitude,  the  equa- 
tions of  the  hourly  motions  have  all  been  rendered  positive,  like 
those  of  the  longitude.  Some  few  new  tables  have  been  com- 
puted for  the  same  purpose.     The  longitude  and  hourly  motion 


248  ASTRONOMY. 

in  longitude  will  very  rarely  differ  from  the  results  of  Burck- 
hardt's  I'ables  more  than  0".5,  and  never  as  much  as  1".  The 
error  of  the  latitude  and  honrly  motion  in  hititiide  will  be  still 
less.  The  other  tables  have  been  taken  from  some  of  the  most 
approved  modern  Astronomical  Works.  (For  the  principles  of 
the  construction  of  the  Tables,  see  Chap.  X.) 

Before  entering  upon  the  explanation  of  each  of  the  tables, 
it  will  be  proper  to  define  a  few  terms  that  will  be  made  use  of 
in  the  sequel. 

The  given  quantity  with  which  a  quantity  is  taken  from  a 
table,  is  called  the  Argu7ne7it. 

The  angular  arguments  are  expressed  in  some  of  the  tables 
according  to  the  sexagesimal  division  of  the  circle.  In  others,  they 
are  given  in  parts  of  the  circle  supposed  to  be  divided  into  a  100, 
a  1000,  or  10000  &c.  parts. 

Tables  are  of  Single  or  Double  Entry,  according  as  they  con- 
tain one  or  two  arguments.  The  Epoch  of  a  table,  is  the  instant 
of  time  for  which  the  quantities  given  by  the  table  are  computed. 
By  the  Epoch  of  a  quantity,  is  meant  the  value  of  the  quan- 
tity found  for  some  chosen  epoch,  from  which  its  value  at 
other  epochs  is  to  be  computed  by  means  of  its  known  rate  of 
variation. 

Table  I,  contains  the  latitudes,  and  longitudes  from  the  me- 
ridian of  Greenwich,  of  various  conspicuous  places  in  different 
parts  of  the  earth.  The  longitudes  serve  to  make  known  the  time 
at  any  one  of  the  places  in  the  table,  when  that  at  any  of  the 
others  is  given.  The  latitude  of  a  place  is  an  important  element 
in  various  astronomical  calculations. 

Table  II,  is  a  table  of  the  Elements  of  the  Orbits  of  the  Planets 
with  their  secular  variations,  and  serves  to  make  known  the  ele- 
ments at  any  given  epoch  different  from  that  of  the  table.  From 
these  the  elliptic  place  of  the  planets  at  the  given  epoch  may  be 
computed. 

Table  III,  is  a  similar  table  for  the  Moon. 

Tables  IV,  V,  VI,  VII,  require  no  explanation. 

Table  VIII,  gives  the  mean  Astronomical  Refractions ;  that  is, 
the  refractions  which  have  place  when  the  barometer  stands  at  30 
inches,  and  the  thermometer  of  Falirenheit  at  50°. 

Table  IX,  contains  the  corrections  of  the  Mean  Refractions  for 


EXPLANATION    OF    THE    TABLES,  249 

+  1  inch  in  the  barometer,  and — 1°  in  the  theniiometer,  from 
which  the  corrections  to  be  appUed,  at  any  observed  height  of  the 
barometer  and  thermometer,  are  easily  derived. 

Table  X,  gives  the  Parallax  of  the  Sun  for  any  given  altitude 
on  a  given  day  of  the  year  ;  for  reducing  a  solar  observation  made 
at  the  surface  of  the  earth  to  what  it  would  have  been,  if  made  at 
the  centre. 

Table  XI,  is  designed  to  make  known  the  Sun's  Semi-diurnal 
Arc,  answering  to  any  given  latitude,  and  to  any  given  declina- 
tion of  the  sun  ;  and  thus  the  time  of  the  sun's  rising  and  set- 
ting, and  the  length  of  the  day. 

Table  XII,  serves  to  make  known  the  value  of  the  Equation  of 
Time,  with  its  essential  sign,  which  is  to  be  applied  to  the  appa- 
rent time  to  convert  it  into  the  mean.  If  the  sign  of  the  equation 
taken  from  the  table  be  changed,  it  will  serve  for  the  conver- 
sion of  mean  time  into  apparent.  This  table  is  constructed  for 
the  year  1840. 

Table  XIII,  is  to  be  used  in  connection  with  Table  XII,  when 
the  given  date  is  in  any  other  year  than  1840.  It  furnishes  the 
Secular  Variation  of  the  Equation  of  Time,  from  which  the  pro- 
portional part  of  its  variation  in  the  interval  between  the  given  date 
and  the  epoch  of  Table  XII  is  easily  derived. 

Table  XIV,  contains  certain  other  Corrections  to  be  applied  to 
the  equation  of  time  taken  from  Table  XII,  when  its  exact  value, 
to  within  a  small  fraction  of  a  second,  is  desired. 

Table  XV,  gives  the  Fraction  of  the  Year,  corresponding  to 
each  date.  This  table  is  useful,  when  quantities  vary  by  Ijnown 
and  uniform  degrees,  in  deducing  their  values  at  any  assumed 
time  from  their  values  at  any  other  time. 

Table  XVI,  is  for  converting  Hours,  Minutes,  and  Seconds 
into  decimal  parts  of  a  Day. 

Table  XVII,  is  for  converting  Minutes  and  Seconds  of  a 
degree  into  the  decimal  division  of  the  same.  It  will  also  serve 
for  the  conversion  of  minutes  and  seconds  of  time  into  decimal 
parts  of  an  hour. 

The  last  two  tables  will  be  found  frequently  useful  in  arith- 
metical operations. 

Table  XVIII,  is  a  table  of  Epochs  of  the  Sun's  Mean  Longi- 
tude, of  the  Longitude  of  the  Perigee,  and  of  the  Arguments  for 
32 


250  ASTRONOMY. 

finding  the  small  equations  of  the  Sun's  place.  They  are  all 
calculated  for  the  first  of  January  of  each  year,  at  mean  noon 
on  the  meridian  of  Greenwich,  Argument  I,  is  the  mean 
longitude  of  the  Moon  minus  that  of  the  Sun  ;  Argument  II,  is 
the  heliocentric  longitude  of  the  Earth ;  Argument  III,  is  the 
heliocentric  longitude  of  Venus  ;  Argument  IV,  is  the  heliocen- 
tric longitude  of  Mars  ;  Argument  V,  is  the  heliocentric  longi- 
tude of  Jupiter ;  Argument  VI,  is  the  mean  anomaly  of  the 
Moon ;  Argument  VII,  is  the  heliocentric  longitude  of  Saturn  ; 
and  Argument  N,  is  the  supplement  of  the  longitude  of  the 
Moon's  Ascending  Node.  Argument  I,  is  for  the  first  part  of  the 
equation  depending  on  the   action  of  the  Moon.     Arguments 

I  and  VI,  are  the  argimients  for  the  remaining  part  of  the  lunar 
equation.  Arguments  II  and  III,  are  for  the  equation  depending 
on  the  action  of  Venus  ;  Arguments  II  and  IV,  for  the  equation 
depending  on  the  action  of  Mars ;  Arguments  II  and  V,  for  the 
equation  depending  on  the  action  of  Jupiter ;  and  Arguments 

II  and  VII,  for  the  equation  depending  on  the  action  of  Saturn. 
Aronment  N,  is  the  argument  for  the  Nutation  in  longitude  :  it 
is  also  the  argument  for  the  Nutation  in  right  ascension,  and  of 
the  obliquity  of  the  ecliptic. 

Table  XIX,  shows  the  Motions  of  the  Sun  and  Perigee,  and  the 
variations  of  the  arguments,  in  the  interval  between  the  beginning 
of  the  year  and  the  first  of  each  month. 

Table  XX,  shows  the  Motions  of  the  Sun  and  Perigee,  and  the 
variations  of  the  arguments,  for  Days  and  Hours. 

Table  XXI,  gives  the  Sun's  Motions  for  Minutes  and  Seconds, 
Tables  XVIII  to  XXI,  make  known  the  mean  longitude  of  the 
Sun  from  the  mean  equinox  at  any  moment  of  time. 

Table  XXII.  Mean  Obliquity  of  the  Ecliptic  for  the  beginning 
of  each  year  contained  in  the  table.  It  is  found  for  any  inter- 
mediate time  by  a  simple  proportion. 

Tables  XXIII  and  XXIV,  furnish  the  Sun's  Hourly  Motion 
and  Semi-diameter. 

Table  XXV,  is  designed  to  make  known  the  Equation  of  the 
Sun's  Centre.  When  the  equation  has  the  negative  sign,  its  sup- 
plement to  12s.  is  taken.  This  is  to  be  added  along  with  the 
other  equations  of  longitude,  and  12s.  are  to  be  subtracted  from 
the  sum.     The  signs  of  the  argument  are  given  both  at  the  head 


EXPLANATION    OF    THE    TABLES.  251 

and  foot  of  the  columns.  The  numbers  in  the  table  are  the 
values  of  the  equation  of  the  centre,  or  of  its  supplement,  dimin- 
ished by  46".  1.  This  constant  is  subtracted  from  each  value,  to 
balance  the  different  quantities  added  to  the  other  equations  of 
the  longitude,  in  order  to  render  them  atfirmative.  The  epoch  of 
this  table  is  the  year  1840. 

Table  XXVI,  a^ives  the  Secular  Variation  of  the  Equation  of 
the  Sun's  Centre,  from  which  the  proportional  part  of  the  varia- 
tion in  the  interval  between  the  given  date  and  the  year  1840, 
may  be  derived. 

Table  XXVII,  is  for  the  Nutation  in  Longitude  and  Right  As- 
cension, and  of  the  Obliquity  of  the  Ecliptic.  The  nutation  in 
longitude  and  in  right  ascension,  serve  to  transfer  the  origin  of 
the  longitude  and  right  ascension  from  the  mean  to  the  true  equi- 
nox. And  the  nutation  of  obliquity  serves  to  change  the  mean 
into  the  true  obliquity. 

Tables  XXVIII  to  XXXIII,  give  the  Equations  of  the  Sun's 
Longitude,  due  respectively  to  the  attractions  of  the  Moon, 
Venus,  Jupiter,  Mars,  and  Saturn. 

Table  XXXIV  is  for  the  variable  part  of  the  Sun's  Aberration. 
The  numbers  have  all  been  rendered  positive  by  the  addition  of 
the  constant  0".3. 

Table  XXXV,  contains  the  Epochs  of  the  Moon's  Mean  Longi- 
tude, and  of  the  Arguments  for  finding  the  equations  which  are 
necessary  in  determining  the  True  Longitude  and  Latitude  of  the 
Moon.  They  are  all  calculated  for  the  first  of  January  of  each 
year,  at  mean  noon  on  the  meridian  of  Greenwich.  The  Argu- 
ment for  the  Evection  is  diminished  by  30' ;  the  Anomaly  by  2° ; 
the  Argument  for  the  Variation  by  9°  ;  and  the  Supplement  of  the 
Node  is  increased  by  7'.  This  is  done  to  balance  the  quantities 
which  are  added  to  the  different  equations  in  order  to  render 
them  affirmative. 

Tables  XXXVI  to  XL,  inclusive,  give  the  Motions  of  the 
Moon,  and  the  variations  of  the  arguments  for  Months,  Days, 
Hours,  Minutes,  and  Seconds  ;  and,  together  with  Table  XXXV, 
are  for  finding  the  Moon's  Mean  Longitude  and  the  Arguments 
at  any  assumed  moment  of  time. 

Tables  XLI  to  LIU,  inclusive,  give  the  various  Equations  of 
the  Moon's  Longitude.    It  is  to  be  observed,  with  respect  to  Table 


252  ASTRONOMY. 

XLI,  that  the  right  hand  figure  of  the  argument  is  supposed  to 
be  dropped.  But  when  the  greatest  attainable  accuracy  is  de- 
sired, it  can  be  retained,  and  a  cypher  conceived  to  be  written 
after  the  numbers  in  the  cohimns  of  Arguments  in  the  table.  In 
Tables  L,  LI,  Lll,  and  LV,  the  degrees  will  be  found  by  referring 
to  the  head  or  foot  of  the  column.     (See  Problem  II,  Note  2). 

Table  LIV,  is  for  the  Nutation  of  the  Moon's  Longitude. 

Tables  LV  to  LIX,  inclusive,  are  for  finding  the  Latitude  of 
the  Moon. 

Tables  LX  to  LXIII,  inclusive,  are  for  the  Equatorial  Parallax 
of  the  Moon. 

Table  LXIV,  furnishes  the  Reductions  of  Parallax  and  of  the 
Latitude  of  a  Place.  The  reduction  of  parallax  is  for  obtaining 
the  parallax  at  any  given  place  from  the  equatorial  parallax.  The 
reduction  of  latitude  is  for  reducing  the  true  latitude  of  a  place,  as 
determined  by  observation,  to  the  corresponding  latitude,  on  the 
supposition  of  the  earth  being  a  sphere.  The  ellipticity  to  which 
the  numbers  in  the  table  correspond  is  ^^-^. 

Tables  LXV  and  LXVI.  Moon's  Semi-diameter,  and  the  Aug- 
mentation of  the  Semi-diameter  depending  on  the  altitude. 

Tables  LVII  to  LXXXV,  inclusive,  are  for  finding  the  Hourly 
Motions  of  the  Moon  in  Longitude  and  Latitude. 

Table  LXXXVI.  Mean  New  Moons,  and  the  Arguments  for  the 
Equations  for  New  and  Full  Moon  in  January.  The  time  of 
mean  new  moon  in  January  of  each  year  has  been  diminished  by 
15  hours,  which  has  been  added  to  the  equations  in  Table 
LXXXIX.  Thus,  4h.  20m.  has  been  added  to  equation  I :  lOh. 
10m.  to  equation  II ;  10m.  to  equation  III ;  and  20m.  to  equa- 
tion IV. 

Tables  LXXXVII  and  LXXXVIII,  are  used  with  the  preceding 
in  finding  the  Approximate  Time  of  Mean  New  or  Full  Moon 
in  any  given  month  of  the  year. 

Table  LXXXIX,  furnishes  the  Equations  for  finding  the 
Approximate  Time  of  New  or  Full  Moon. 

Table  XC,  contains  the  Mean  Right  Ascensions  and  Declina- 
tions of  50  principal  Fixed  Stars,  for  the  beginning  of  the  year 
1840,  with  their  Annual  Variations. 

Table  XCI,  is  for  finding  the  Aberration  and  Nutation  of  the 
Stars  in  the  preceding  catalogue. 


EXPLANATION    OF    THE    TABLES.  253 

Table  XCII,  contains  the  Mean  Longitudes  and  Latitudes  of 
some  of  the  principal  Fixed  Stars,  for  the  beginning  of  the  year 
1840,  with  their  Annual  Variations. 

Tables  XCIII,  XCIV,  XCV.  Second,  Third,  and  Fourth  Dif- 
ferences, These  tables  are  useful  for  finding,  from  the  Nautical 
Almanac,  the  moon's  longitude  or  latitude  for  any  time. between 
noon  and  midnight. 

Table  XCVI.  Logistical  Logarithms.  This  table  is  conve- 
nient in  working  proportions  when  the  terms  are  minutes  and 
seconds,  or  degrees  and  minutes,  or  hours  and  minutes, — espe- 
cially when  the  first  term  is  Ih.  or  60m. 

Tojitid  the  logistical  logarithm  of  a  number  composed  of 
minutes  and  seconds,  or  degrees  and  minutes  of  an  arc  ;  or  of 
minutes  and  seconds,  or  hours  and  mi?iutes  of  time. 

1.  If  the  number  consists  of  minutes  and  seconds,  at  tlie  top 
or  bottom  of  the  table  seek  for  the  minutes,  and  in  the  same 
column  opposite  the  seconds  in  the  left-hand  column  will  be 
found  the  logfistical  los^arithm. 

2.  If  the  number  is  composed  of  hours  and  minutes,  the  hours 
must  be  used  as  if  they  were  minutes,  and  the  minutes  as  if 
they  were  seconds. 

3.  If  the  number  is  composed  of  degrees  and  minutes,  the  de- 
grees must  be  used  as  if  they  were  minutes,  and  the  minutes  as 
if  they  were  seconds. 

Tofiiid  the  logistical  logarithm,  of  a  number  less  than  3600. 

Seek  in  the  second  line  of  the  table  from  the  top  the  number 
next  less  than  the  given  number,  and  the  remainder,  or  the  com- 
plement to  the  given  number,  in  the  first  column  on  the  left : 
then,  in  the  column  of  the  first  number,  and  opposite  the  com- 
plement, will  be  found  the  logistical  logarithm  of  the  sum. 
Thus,  to  obtain  the  logarithm  of  1531,  we  seek  for  tlie  column 
of  1500,  and  opposite  31  we  find  3713. 


4-- 
254  ASTRONOMY. 

PROBLEM    I. 

To  work,  hy  logistical  logarithms,  a  proportion  the  terms  of 

which  are  degrees  and  minntes,  or  tninutes  and  seconds  of 

an  arc ;  or  hours  and  minutes,  or  minutes  and  seconds  of 

time. 

With  the  degrees  or  minutes  at  the  top,  and  minutes  or  seconds 
at  the  side,  or  if  a  term  consists  of  hours  and  minutes,  or  minutes 
and  seconds,  with  the  hours  or  minutes  at  the  top,  and  minutes 
or  seconds  at  the  side,  take  from  Table  XCVI  the  logistical  loga- 
rithms of  the  three  given  terms  ;  add  together  the  logistical  loga- 
rithms of  the  second  and  third  terms  and  the  arithmetical  comple- 
ment of  that  of  the  first  term,  rejecting  10  from  the  index.*  The 
result  will  be  the  logistical  logarithm  of  the  fourth  term,  with 
which  take  it  from  the  table. 

Note  1.  The  logistical  logarithm  of  60'  is  0. 

Note  2,  If  the  second  or  third  term  contains  tenths  of  seconds, 
(or  tenths  of  minutes,  when  it  consists  of  degrees  and  minutes), 
and  is  less  than  6',  or  6°,  multiply  it  by  10,  and  employ  the  loga- 
rithm of  the  product  in  place  of  that  of  the  term  itself  The 
result  obtained  by  the  table  divided  by  10  will  be  the  fourth  term  of 
the  proportion,  and  will  be  exact  to  tenths. 

Note  3.  If  none  of  the  terms  contain  tenths  of  minutes  or 
seconds,  and  it  is  desired  to  obtain  a  result  exact  to  tenths, 
diminish  the  index  of  the  logistical  of  the  fourth  term  by  1,  and 
cut  off  the  right-hand  figure  of  the  number  found  from  the 
table,  for  tenths. 

Exam.  1.  When  the  moon's  hourly  motion  is  30'  12",  what  is 
its  motion  in  16m.  24s.  1 

As  60m.  -    -    -    -    0 

:  30'  12"    -    -    -  2981 

:  :  16m.  24s.  -    -    -  5633 


8' 15"    -   -   -  8614 


*  Instead  of  adding  the  arithmetical  complement  of  the  logarithm  of  the  first 
term,  the  logarithm  itself  may  be  subtracted  from  the  sum  of  the  logarithms  of 
the  other  two  terms. 


PROB.   11.,  TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.      255 

2.  If  the  moon's  declination  change  1°  31'  in  12  hours,  what 
will  be  the  change  in  7h.  42m.  ? 

As  12h.  -         -         Ar.  Co.  9.3010 

:    I03I'         -        -        -     1.5973 

::    7h.  42m.     -         -         -       8917 


:    0°  58'         -        -        -     1.7900 
3.  When  the  moon's  hourly  motion  in  latitude  is  2'  26".8, 
what  is  its  motion  in  36m.  22s.  ? 
2'  26".8 
60 


146".8 
10 

1468 


As    60m.    - 

0 

:    1468"   - 

-  3896 

::    36m.  22s. 

-  2174 

:     890"     -        -  6070 

Ans.  1'  29".0. 

4.  When  the  sun's  hourly  motion  in   longitude  is  2'  28", 
what  is  its  motion  49m.  Us.?  Ans.   2'  1". 

5.  If  the  sun's  declination  changes  16'  33"  in  24  hours,  what 
will  be  the  change  in  14h.  18m.  ?  Ans.  9'  52". 

6.  If  the  moon's  declination  change  54".7  in  one  hour,  what 
will  be  the  change  in  52m.  18s.  ?  Ans.  47". 7. 

PROBLEM    II. 

To  take  f torn  a  table  the  quantity  corresponding  to  a  given 

value  of  the  argument^  or  to  given  values  of  the  arguments 

of  the  table. 

Case  1.  When  quantities  are  given  in  the  table  for  each  sign 
and  degree  of  the  argument. 

With  the  signs  of  the  given  argument  at  the  top  or  bottom,  and 
the  degrees  at  the  side,  (at  the  left  side,  if  the  signs  are  found  at 
the  top ;  at  the  right  side,  if  they  are  found  at  the  bottom), 
take  out  the  corresponding  quantity.  Also  take  the  differ- 
ence between  this  quantity  and  the  next  following  one  in  the 
table,  and  say,  60' :  this  difference  :  :  odd  minutes  and  seconds  of 


256  ASTRONOMY. 

given  argument :  a  fourth  term.  This  fourth  term,  added  to  the 
quantity  taken  out,  when  the  quantities  in  the  table  are  increas- 
ing, but  subtracted  when  they  are  decreasing,  will  give  the  re- 
quired quantity. 

Note  1.  When  the  quantities  change  but  little  from  degree  to 
degree,  the  required  quantity  may  frequently  be  estimated  without 
the  trouble  of  making  a  proportion. 

Note  2.  In  some  of  the  tables  the  degrees  or  signs  of  the  quan- 
tity sought,  are  to  be  had  by  referring  to  the  head  or  foot  of  the 
column  in  which  the  minutes  and  seconds  are  found.  (See 
Tables  L,  LI,  LII,  and  LY.)  The  degrees  there  found  are  to  be 
taken,  if  no  horizontal  mark  intervenes ;  otherwise,  they  are  to  be 
increased  or  diminished  by  1°,  or  2°,  according  as  one  or  two 
marks  intervene.  They  are  to  be  increased,  or  diminished,  ac- 
cording as  their  number  is  less  or  greater  than  the  number  of 
degrees  at  the  other  end  of  the  column. 

Note  3.  If,  as  is  the  case  with  some  of  the  tables,  the  quantities 
in  the  table  have  an  algebraic  sign  prefixed  to  them,  neglect  the 
consideration  of  the  sign  in  determining  the  correction  to  be  ap- 
plied to  the  quantity  first  taken  out,  and  proceed  according  to  the 
rule  above  given.  The  result  will  have  the  sign  of  the  quantity 
first  taken  out.  It  is  to  be  observed,  however,  that  if  the  two 
consecutive  quantities  chance  to  have  opposite  signs,  their  nu- 
merical sum  is  to  be  taken  instead  of  their  diflerence  ;  also  that 
the  quantity  sought  will,  in  every  such  instance,  be  the  numerical 
difference  betv/een  the  correction  and  the  quantity  first  taken  out, 
and,  according  as  the  correction  is  less  or  greater  than  this  quan- 
tity, is  to  be  affected  with  the  same  or  the  opposite  sign. 

Exam.  1.  Given  the  argument  7^  6°  24'  36",  to  find  the  cor- 
responding quantity  in  Table  L. 

7^  6^  gives  0°  43'  17".4. 

The  difference  between  0°  43'  17".4  and  the  next  following 
quantity  in  the  table  is  1'  7".3. 

60'  :  1'  7".3  :  :  24'  36"  :  27".6.* 


*  The  student  can  work  the  proportion  either  by  common  arithmetic,  or  by 
logistical  logarithms,  as  he  may  prefer.  In  working  this  and  all  similar  propor- 
tions by  tlie  arithmetical  method,  the  seconds  of  the  argument  may  be  converted 
into  the  equivalent  decimal  part  of  a  minute  by  means  of  Table  XVII.  It  will  be 
sufficient  to  take  the  fraction  to  the  nearest  tenth. 


PROB.  11,  TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.   257 

From     0°  43'  17".4 
Take  27  .6 


0    42   49  .8 
2.  Given  the  argument  2«18°  41'  20",  to  find  the  correspond- 
ing quantity  in  Table  XXV. 

2^- 18^  gives  1°  52'  32".5. 
The  diflference  between  1°  52'  32".5  and  the  next  following 
quantity  in  the  table  is  21".8. 

60'  :  2r'.8  :  :  41'  20"  :  15".0. 

To         1°  52'  32".5 
Add  15  .0 


1     52   47  .5 
3.  Given  the  argument  9^-  2°  13'  33",  to  find  the  correspond- 
ing quantity  in  Table  XII. 

9'-  2°  gives  29.8s. 
The  arithmetical  sum  of  29.8s.  and  the  next  following  quan- 
tity in  the  table  is  30.4s. 

60' :  30.4s.  :  :  13°  33'  :  6.9s. 
From     29.8s. 
Take       6.9 


22.9s. 

4.  Given  the  argument  5^-  8°  14'  52",  to  find  the  correspond- 
ing quantity  in  Table  LII.  Ans.  12'  36".0. 

5.  Giv'en  the  argument  11^- 11°  23'  10",  to  find  the  correspond- 
ing quantity  in  Table  LVI.  Ans.  11'  48". 0. 

6.  Given  the  argument  0^-  26°  20',  to  find  the  corresponding 
quantity  in  Table  XII.  Ans.  —  4P.0. 

Case  2.  When  the  argument  changes  in  the  table  by  more 
or  less  than  1°  ;  or  when  it  is  given  in  lower  denominations 
than  signs. 

Take  out  of  the  table  the  quantity  answering  to  the  number 
in  the  column  of  arguments  next  less  than  the  given  argument. 
Take  the  difference  between  this  quantity  and  the  next  follow- 
ing one,  and  also  the  difference  of  the  consecutive  values  of  the 
argument  inserted  in  the  table,  and  say.  difference  of  argu- 
ments :  difference  of  quantities  :  :  excess  of  the  given  argument 
over  the  value  next  less  in  the  table  :  a  fourth  term.  This 
33 


258  ASTRONOMY. 

fourth  term  applied  to  the  quantity  first  taken  out,  accordins:  to 
the  rule  given  in  the  preceding  case,  will  give  the  quantity 
sought. 

Note  3.  In  some  of  the  tables  the  columns  entitled  Diif.  are 
made  up  of  the  differences  answering  to  a  diiference  of  10'  in 
the  arofument.  In  obtaining  quantities  from  these  tables,  it  will 
be  found  more  convenient  to  take  for  the  first  and  second  terms 
of  the  proportion,  respectively,  10',  and  the  difference  furnished 
by  the  table,  and  work  the  proportion  by  the  arithmetical 
method.     (See  note  at  bottom  of  page  256). 

Exam.  1.  Given  the  argument  0^-  24^^  42'  15",  to  find  the  cor- 
responding quantity  in  Table  LI. 

0^  24"  30'  gives  9°  47'  14".3.     ' 

The  difference  between  9^  47'  14". 3  and  the  next  following 
quantity  =  3  X  63".0  =  189". 0.  The  argument  changes  by  30'. 
And  the  excess  of  0^-  24°  42'  15"  over  0^-  24°  30'  is  12'  15". 
Thus, 

30'  :  189  ".0  :  :  12'  15"  :  77".2. 
But  the  correction  may  be  found  more  readily  by  the  following 

proportion, 

10'  :  63  ".0  :  :  12.25  :  77".2. 

To         9"  47    14".3 
Add  77  .2 


9°  48'  31  .5 
2.  Given  the  argument  1°  12',  to  find  the  corresponding  quan- 
tity in  Table  YIII. 

1°  10'  gives  23'  13", 

and  5'  :  27"  :  :  2'  :  11"  the  correction. 

From  23'  13" 

Take  11 


23     2 

3.  Given  the  argument  6*-  6°  7'  23",  to  find  the  correspond- 
ing quantity  in  Table  LV.  Ans.  90°  20'  53".5. 

4.  Given  the   argument  49°  27',  to  find  the   corresponding 
quantity  in  Table  LXIV.  Ans.  11'  19".8. 

Case  3.    When  the  argiiinejit  is  given  in  the  table  in  hun- 
dredth, thousandth,  or  ten  thousandth  parts  of  a  circle. 

The  required  quantity  can  be  found  in  this  case  by  the  same 


PROB.  II,  TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.   259 

Tule  as  in  the  preceding  ;  but  it  can  be  had  more  expeditiously 
by  observing  the  following  rules.  If  the  argument  varies  by 
10,  multiply  the  difference  of  the  quantities  between  which  the 
required  quantity  lies  by  the  excess  of  the  given  argument  over 
the  next  less  value  in  the  table,  and  remove  the  decimal  point 
one  figure  to  the  left ;  the  result  will  be  the  correction  to  be 
applied  to  the  quantity  taken  out  of  the  table.  The  same  rule 
will  apply  in  taking  quantities  from  tables  in  which  the  differ- 
ences, answering  to  a  change  of  10  in  the  argument,  are  sfiven, 
although  the  argument  should  actually  change  by  50  or  100. 
If  the  argument  changes  by  100,  multiply  as  above,  and  remove 
the  decimal  point  two  figures  to  the  left.  When  the  common 
difference  of  the  arguments  is  5,  proceed  as  if  it  were  10,  and 
double  the  result.  In  like  manner,  when  the  common  differ- 
ence is  50,  proceed  as  if  it  were  100,  and  double  the  result. 

Exam.  1.  Given  the  argument  973,  to  find  the  corresponding 
quantity  in  Table  XLV,  column  headed  13. 
970  gives  23".5. 

The  diflference  is  1".2,  and  the  excess  3. 

1".2  From    23".5 

3  Take  .4 


Corr.  .36  23  .1 

2.  Given  the  argument  4834,  to  find  the  corresponding  quan- 
tity in  Table  XLII,  column  headed  5. 

4800  gives  2'  3".7. 
The  difference  is  6".8,  and  the  excess  34. 
6".8 
34 

From    2'  3".7 

2.312        -        -        -     Take         2  .3 


2    1  .4 

3.  Given  the  argument  5444,  to  find  the  corresponding  quan- 
tity in  Table  XLL  Ans.  15'  37".7. 

4.  Given  the  argument  4225,  to  find  the  corresponding  quan- 
tity in  Table  XLIII,  column  headed  8.  Ans.  0'  47".2. 

Case  4.    Whe7i  the  table  is  one  of  double  entry ^  or  quantities 
are  taken/ram  it  by  means  of  two  arguments. 

Take  out  of  the  table  the  quantity  answering  to  the  values  of 


260 


ASTRONOMY. 


the  argfiiments  of  the  tahle  next  less  than  the  given  values  ;  and 
find  the  respective  corrections  to  be  applied  to  it,  due  to  the 
excess  of  the  given  value  of  each  argument  over  the  next  less 
value  in  the  table,  by  the  general  rule  given  in  the  preceding 
case.  These  corrections  are  to  be  added  to  the  quantity  taken 
out,  or  subtracted  from  it,  according  as  the  quantities  increase 
or  decrease  with  the  arguments. 

Note  1.  If  the  tenths  of  seconds  be  omitted,  the  corrections 
above  mentioned  can  be  estimated,  without  the  trouble  of  sta- 
ting a  proportion,  or  performing  multiplications. 

Note  2.  The  rule  above  given  may,  in  some  rare  instances, 
give  a  result  differing  a  few  tenths  of  a  second  from  the  truth. 
The  following  rule  will  furnish  more  exact  results.  Find  the 
quantities  corresponding,  respectively,  to  the  value  of  the  argu- 
ment at  the  top  next  less  than  its  given  value,  and  the  other 
given  argument,  and  to  the  value  next  greater  and  the  other 
given  argument.  Take  the  difference  of  the  quantities  found, 
and  also  the  difference  of  the  corresponding  arguments  at  top, 
and  say,  difference  of  arguments  :  difference  of  quantities  : :  ex- 
cess of  given  value  of  the  argument  at  the  top  over  its  next  less 
value  in  the  table  :  a  fourth  term.  This  fourth  term  added  to 
the  quantity  first  found,  if  it  is  less  than  the  other,  but  subtracted 
from  it,  if  it  is  greater,  will  give  the  required  quantity.  The 
error  of  the  first  rule  may  be  diminished  without  any  extra  calcu- 
lation, by  attending  to  the  differences  of  the  quantities  answering 
to  the  value  of  the  argument  at  the  side  next  greater  than  its 
given  value,  and  the  values  of  the  other  argument,  between 
which  its  given  value  lies. 

Exam.  1.  Given  the  argument  64  at  the  top  and  77  at  the  side, 
to  find  the  corresponding  quantity  in  Table  LXXXI. 
50^and  70  give  47".7. 

The  difference  between  47". 7  and  the  next  quantity  below  it 
is  1".4.  The  excess  of  77  over  70  is  7,  and  the  argument  at  the 
side  changes  by  10. 

1".4 
7 

From     47".7 

Corr.  due  excess  7,      .98,  or  1  ".0.       Take       1  .0 


Quantity  corresponding  to  50  and  77,      46  .7 


PROB.  II,  TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.    261 

The  difference  between  47".7  and  the  adjacent  quantity  in  the 
next  column  on  the  right  is  3". 3.     The  excess  of  64  over  50  is 
14,  and  the  argument  at  the  top  changes  by  50. 
3".3 
14 


.462 
2 


From    46".7 

Corr.  due  excess  14,     .924  Take       0  .9 


45  .8 

2.  Given  the  argument  223  at  the  top  and  448  at  the  side,  to 

lind  the  corresponding  quantity  in  Table  XXX. 

220  and  440  give  16".0. 

The  difference  between  16".0  and  the  quantity  next  below  it 

is  2".2. 

2".2 

8 


2  )  1.76 

From     16".0 

Corr.  for  excess  8,  .88,  or  0".9.       Take       0  .9 


Q^uantity  corresponding  to  220  and  448,    15  .1 
The  difference  between  16".0  and  the  adjacent  quantity  in  the 
next  column  on  the  right  is  0".7. 
0".7 

3 

To       15".l 


Corr.  for  excess  3,  .21  Add  .2 


15  .3 

3.  Given  the  argument  472  at  the  top  and  786  at  the  side,  to 
find  the  corresponding  quantity  in  Table  XXXI. 

Ans.  9".7. 

4.  Given  the  argument  620  at  the  top  and  367  at  the  side,  to 
find  the  corresponding  quantity  in  Table  LXXXI. 

Ans.  55".2. 

5.  Given  the  argument  348  at  the  top  and  932  at  the  side,  to 
find  (by  the  rule  given  in  Note  2)  the  corresponding  quantity  in 
Table  XXXII.  Ans.  15".4.    _ 


262  ASTRONOMY. 


PROBLEM    III. 

To  convert  Degrees,  Minutes,  and  Seconds  of  the  Equator  into 

Time. 
Multiply  the  quantity  by  4,  and  call  the  product  of  the  seconds 
thirds  ;  of  the  minutes,  seconds  ;  and  of  the  degrees,  minutes. 
Exam  1.  Convert  83°  11'  52"  into  time. 
83°  11'    52" 
4 


5h   som.  47s.  28'" 
2.  Convert  34°  57'  46"  into  time. 

Ans.  2h.  19m.  51sec.  4'". 


PROBLEM    IV. 

To  convert  Time  into  Degrees,  Minnies,  and.  Seconds. 

Reduce  the  hours  and  minutes,  to  minutes  ;  divide  by  4,  and 

call  the  quotient  of  the  minutes,  degrees  ;  of  the  seconds,  minutes  ; 

and  multiply  the  remainder  by  15,  for  the  seconds. 

Exam.  1.  Convert  7h.  9m.  34sec,  into  degrees,  &.c. 
7h.  gm.  34s. 

60 


4 )  429     34 


107°  24'  30" 
2.  Convert  llh.  24m.  45s.  into  degrees,  <fec. 

Ans.  171°iri5". 


PROBLEM    V. 

The  Longitudes  of  two  Places,  and  the  Time  at  one  of  them 
being  given,  to  find  the  corresponding  time  at  the  other. 
"When  the  given  time  is  in  the  morning,  change  it  to  astronom- 
ical time,  by  adding  12  hours,  and  diminishing  the  number  of  the 


PROC.  V,  TO  REDUCE  TIME  FROM  ONE  PLACE  TO  ANOTHER.    263 

day  by  a  unit.     When  the  given  time  is  in  the  evening,  it  is 
already  in  astronomical  time. 

Find  the  difference  of  longitude  of  the  two  places,  by  taking  the 
numerical  difference  of  their  longitudes,  when  these  are  of  the 
same  name;  that  is,  both  east  or  both  west;  and  the  sum,  when 
they  are  of  different  names  ;  that  is,  one  west  and  the  other  east. 
When  one  of  the  places  is  Greenwich,  the  longitude  of  the  other 
is  the  difference  of  longitude. 

Then,  if  the  place  at  which  the  time  is  required,  is  to  the  east 
of  the  other  place,  add  the  difference  of  longitude,  in  time,  to  the 
given  time  ;  but,  if  it  is  to  the  icest,  subtract  the  difference  of  lon- 
gitude, from  the  given  time.  The  sum  or  remainder  will  be  the 
requir.^d  time. 

Note.  The  longitudes  of  the  places  mentioned  in  the  following 
examples,  are  given  in  Table  I. 

Exam.  1.  When  it  is  October  25th,  3h.  13m.  22sec.  A.  M.,  at 
Greenwich,  what  is  the  time,  as  reckonrd  at  New  York? 
Time  at  Greenwich,  October,    24'-  15'^-  IS"- 22^- 
Diff.ofLona.    -        -        -  4     56      4 


Time  at  New  York    -        -      24     10     17     18  P.M. 
2.  When  it  is  June  9;h,  5h.  25m.  lOsec.  P.  M.  at  Washington, 
what  is  the  corresponding  time  at  Greenwich? 

Tinje  at  Wcishington,  June,  9^^-   5^   25">-  10^- 

DiffofLong.      -         -        -  5       8      7 


Time  at  Greenwich    -         -  9  10     33     17  P.  M. 

3.  When  it  is  January  15th,  2h.  44m.  23sec.  P.  M.  at  Paris, 
what  is  the  time  at  Philadelphia? 

Longitude  of  Paris,     -        -        -  O'^-  9"  21^6    E 
Do.         of  Philadelphia,  -        -  5     0     44        W. 


5   10       5.6 


Time  at  Paris,  January,         -     15'^-  2^-  44'"-  23^- 
Diff.  of  Long,         -        -        -  5     10      6 

Time  at  Philadelphia,    -        -     14  21     34     17 
Or  Jarmary  15th,  9h.  34m.  17sec.  A.  M. 


264  ASTllONOMY. 

4.  When  it  is  March  31st,  8h.  4m.  21sec.  P.  M.  at  New  Haven, 
what  is  the  corresponding  time  at  Berhn  ? 

Ans.  April  1st,  Ih.  49m.  48sec.  A.  M. 

5.  When  it  is  August  10th,  lOh.  32m.  14sec.  A.  M.  at  Boston, 
what  is  the  time  at  New  Orleans  ? 

Ans.  Aug.  10th,  9h.  16m.  3sec.  A.  M. 

6.  When   it  is  noon  of  the  23d  of  December  at  Greenwich, 
what  is  the  time  at  New  York  ? 

Ans.  Dec.  23d,  7h.  3m.  55sec.  A.  M. 


PROBLEM    VI. 

The  Apparent   Time  being  given,  to  find  the  corresponding 

Mean   Time ;  or  the  Mean   Time  being  given,  to  find  the 

Apparent. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re- 
duce it  to  that  meridian  by  the  last  problem.  Then  find  by  the 
tables  the  sun's  mean  longitude  corresponding  to  this  time.  Thus, 
from  Table  XVIII  take  out  the  longitude  answering  to  the  given 
year,  and  from  Tables  XIX,  XX,  and  XXI  take  out  the  motions  in 
longitude  for  the  given  month,  days,  hours  and  minutes,  neglect- 
ing the  seconds.  The  sum  of  the  quantities  taken  from  the  tables, 
rejecting  12  signs,  when  it  exceeds  that  quantity,  will  be  the  sun's 
mean  longitude  for  the  given  time. 

With  the  sun's  mean  longitude,  thus  found,  take  the  Equation 
of  Time  from  Table  XII.  Then,  when  Apparent  Time  is  given  to 
find  the  Mean,  apply  the  equation  with  the  sign  it  has  in  the  table  ; 
but  when  Mean  Time  is  given  to  find  the  Apparent,  apply  it  with 
the  contrary  sign ;  the  result  will  be  the  Mean  or  Apparent  Time 
required. 

This  rule  will  be  sufficiently  exact  for  ordinary  purposes,  for 
several  years  before  and  after  the  year  1840.  When  the  given 
date  is  a  number  of  years  distant  from  this  epoch,  take  also  with 
the  sun's  mean  longitude  the  Secular  Variation  of  the  Equation  of 
Time  from  Table  XIII,  and  find  by  simple  proportion  the  variation 
in  the  interval  between  the  given  year  and  1840.  The  result, 
applied  to  the  equation  of  time  taken  from  Table  XII,  according  to 


PROB,  VI.    TO    CONVERT    APPARENT    INTO  MEAN  TIME.       265 

its  sign,  if  the  given  time  is  subsequent  to  the  year  1 840,  but  with 
the  opposite  sign,  if  it  is  prior  to  1840,  will  give  the  equation  of 
time  at  the  given  date,  which  apply  to  the  given  time  as  above 
directed. 

Note  1.  When  the  exact  mean  or  apparent  time  to  within  a 
small  fraction  of  a  second  is  demanded,  take  the  numbers  in  the 
cokimns  entitled  I,  II,  III,  IV,  V,  N,  in  Tables  XVII,  XIX,  XX, 
XXI,  ansvi'^ering  respectively  to  the  year,  months,  days,  hours,  and 
minutes  of  the  given  time.  With  the  respective  sums  of  the 
numbers  taken  from  each  column,  as  arguments,  enter  Table  XIV, 
and  take  out  the  corresponding  quantities.  These  quantities  add- 
ed to  the  equation  of  time  as  given  by  Tables  XII  and  XIII,  and 
the  constant  3.0s.  subtracted,  will  give  the  true  Equation  of 
Time,  if  the  given  time  is  Mean  Time.  When  Apparent  Time  is 
given,  it  will  be  farther  necessary  to  correct  the  equation  of  time  as 
given  by  the  tables,  by  stating  the  proportion,  24  hours  :  change 
of  equation  for  1°  of  longitude  :  :  equation  of  time  :  correction. 

Note  2.  The  Equation  of  Time  is  given  in  the  Nautical  Alma- 
nac for  each  day  of  the  year,  at  apparent,  and  also  at  mean  noon, 
on  the  meridiem  of  Greenwich,  and  can  easily  be  found  for  any 
intermediate  time  by  proportion.  Directions  for  applying  it  to 
the  given  time  are  placed  at  the  head  of  the  column.  The  Equa- 
tion is  given  on  the  first  and  second  pages  of  each  month. 

Exam.  1.  On  the  16th  of  July,  1840,  when  it  is9h.  35m.  22s.  P. 
M.  mean  time  at  New  York,  what  is  the  apparent  time  at  the 
same  place  ? 

Time  at  New  York,  July  1840,  -  16''   9^-  35'"-  22«- 
Diff.  of  Long.   -        -        -        -  4    56       4 


Time  at  Greenwich,  July  1840,  16  14     31     26 

M.  Long. 

1840 9=-  10°  12'  49" 

July 5   29    23   16 

16d. 14    47     5 

14h. 34   30 

31m. 1    16 

M.Long.         -        -        -        -  3   24    58   56 
34 


266 


ASTRONOMY. 


The  equation  of  time  in  Table  XII,  corresponding  to  3'-  24°  58' 
56",  is  +  5""   44^- 

Mean  Time  at  New  York,  July  1840, 16  '■  9^-  35'"  22^ 
Equation  of  time,  sign  changed,         -  —  5     44 

Apparent  Time,  16     9     29     38  P.  M. 

2.  On  the  9th  of  May,  1842,  when  it  is  4h.  15m.  21sec.  A.  M, 
apparent  time  at  New  York,  what  is  the  mean  time  at  the  same 
place,  and  also  at  Green wicii  ? 

Time  at  New  York,  May  1842,  8'^-  le^^-  15"'-  2V 
Diff.  ofLong.         -        -        -  4    56      4 

Time  at  Greenwich,        -        -  8     21     11     25 

M.  Long. 
1842     -        -     9^-  10°  43"^  18'- 
May     -        -     3     28    16     40 
8d.     -        -  6   53     58 

21  h.    -        -  51     45 

11m.    -        -  27 


M.Long.      -     1     16   46       8.  Equa.  of  time,  —3m.  45s. 
Apparent  Time  at  Greenwich,  May,  1842,         8^-  21^-  ll-^^S'- 
Equation  of  Time, —  3     45 


Mean  Time  at  Greenwich, 
Diff.  of  Long.    - 


8     21       7     40 
4    56      4 

8     16     11     36 


Mean  Time  at  New  York, 

Or,  May  9th,  4h.  11m.  36s.  A.  M. 

3.  On  the  3d  of  February,  1855,  when  it  is  2h.  43m.  36s.  appa- 
rent time  at  Greenwich,  what  is  the  exact  mean  time  at  the  same 
place  ? 

Appar.  Time  at  Greenwich,  Feb.,  1855,  3d.  2h.  43m.  36s. 


1S.55  .  . 
Feb. 

3d.  .  . 

2h.  .  . 
43  in.  .  . 

M.  honfr. 

I. 

II. 

in. 

806 

1.38 

9 

IV. 

8^9 

45 

3 

V. 

866 
7 
0 

873 

N. 

863 
5 
0 

868 

9'  10^  34  30 

1   0  33  18 

1  5d  17 

4  .56 

1  46 

433 

47 

68 

3 

279 

85 
5 

10  13  12  47 

.551 

369 

953 

937 

PROB.  VII.  TO  CORRECT  AN  OBS.  ALT.  FOR  REFRACTION.   267 

Appar.  Time  at  Greenwich,  Feb.  1855,  3^-  2^-  43="  36^- 

Equation  of  time  by  Table  XII,  -         -  +14       8.6 
lOOyrs.  :  13s.  ( Sec.  Var.  Table  XIII) 

:  :  15yrs.  :  1.9s.  -         -         -         -  _  1.9 


Approx.  Mean  Time  at  Greenwich,     -    3     2     57     42.7 
24h.  :  6s.  (change  of  equa.  for  1°  of 

long.)  ::  14m.  :  0.1s.  -         -         -         -  +0.1 

II.  III. 0.8 

II.  IV. 1.0 

11.  V. 0.4 

I. 0.3 

N. 0.1 

Constant. —3.0 


Mean  Time  at  Greenwich  -     3     2    57    42.4 

4.  On  the  18th  of  November,  1841,  when  it  is  2h.  12in.  26sec. 
A.  M.  mean  time  at  Greenwich,  what  is  the  apparent  time  at 
Philadelphia?  Ans.  Nov.  17th,  9h.  26m.  24s.  P.M. 

5.  On  the  2d  of  February,  1839,  when  it  is  6h.  32m.  35sec. 
P.  M.  apparent  time  at  New  Haven,  what  is  the  mean  time  at  the 
same  place?  Ans.  6h.  46m.  38s.  P.  M. 

6.  On  the  23d  of  September,  1850,  when  it  is  9h.  10m.  12sec. 
mean  time  at  Boston,  what  is  the  exact  apparent  time  at  the  same 
place?  Ans.  9h.  8m.  1.0s. 


PROBLEM   VII. 

To  correct  the    Observed  Altitude  of  a  Heavenly  Body  for 

Refraction. 

With  the  given  altitude  take  the  corresponding  refraction  from 
Table  VIII.  Subtract  the  refraction  from  the  given  altitude, 
and  the  result  will  be  the  true  altitude  of  the  body  at  the  given 
station. 

This  rule  will  give  exact  results  if  the'  barometer  stands  at  30 
inches,  and  Fahrenheit's  thermometer  at  50",  and  results  sufficient- 
ly exact  for  ordinary  purposes  in  any  state  of  the  atmosphere. 
When  there  is  occasion  for  greater  precision,  take  from  Table  IX 


268  ASTRONOMY. 

the  corrections  for  +  1  inch  in  the  height  of  the  barometer,  and 
—  1°  in  the  height  of  Fahrenheit's  thermometer,  and  compute  the 
corrections  for  the  difference  between  the  observed  height  of  the 
barometer  and  30  in.  and  for  the  difference  between  the  ob- 
served height  of  the  thermometer  and  50".  Add  these  to  the  mean 
refraction  taken  from  Table  VIII,  if  the  barometer  stands  higher 
than  30  in.  and  the  thermometer  lower  than  BO'^;  but  in  the 
opposite  case,  subtract  them,  and  the  result  will  be  the  true  refrac- 
tion, which  subtract  from  the  observed  altitude. 

Exam.  1.  The  observed  altitude  of  the  sun  being  32°  10'  25", 
what  is  its  true  altitude  at  the  place  of  observation  ? 

Observed  alt. 32°  10'  25" 

Refraction  (Table  VIII)    -         -         -  —  1    32 


True  alt.  at  the  station      -         -         -  32°     8    53 

2.  The  observed  altitude  of  Sirius  being  20°  42'  11",  the  ba- 
rometer 29.5  inches,  and  the  thermometer  of  Fahrenheit  70°,  re- 
quired the  true  altitude  at  the  place  of  observation.  The  differ- 
ence between  29.5  inches  and  30  inches  is  0.5  inches,  and  the' 
difference  between  70°  and  50°  is  20°. 
Obs.  alt.      -      20°  42'  11  ".0 


Refrac, (Table  VIII),  2'  33".0;  Bar.-l-lin.,5".12;  ther.— 1°,  0".310 
Corr.for— 0.5in.bar.   —  2  .6  5  20 

Corr.for-f  20°  ther.     —6.2  

2.560  6.20 


True  refrac.      -         2  24  .2 


True  alt.      -    20     39   46  .8 

3.  The  observed  altitude  of  the  moon  on  the  11th  of  April,  1838, 
being  14°  17'  20",  required  the  true  altitude  at  the  place  of  obser- 
vation. Ans.  14°  13'  35". 

4.  Let  the  observed  altitude  of  Aldebaran  be  48°  35'  52",  the 
barometer  at  the  same  time  standing  at  30.7  inches,  and  the  ther- 
mometer at  42°,  required  the  true  altitude. 

Ans.  48°  34'  58".8. 


PROB.  VIII.  TO  DEDUCE  THE  TRUE  FROM  THE  APPAR.  ALT.      269 


PROBLEM   VIII. 

The  Apparent  Altitude  of  a  Heavenly  Body  being  given,  to 
find,  its  True  Altitude. 
Correct  the  observed  altitude  for  refraction  by  the  foregoing 
problem.     Then, 

1.  If  the  sun  is  the  body  whose  altitude  is  taken,  find  its  paral- 
lax in  altitude  by  Table  X,  and  add  it  to  the  observed  altitude 
corrected  for  refraction.  The  reiult  will  be  the  true  altitude, 
sought. 

2.  If  it  is  the  altitude  of  the  moon  that  is  taken,  and  the  hori- 
zontal parallax  at  the  time  of  the  observation  is  known,  find  the 
parallax  in  altitude  by  the  following  formula : 

log.  sin  (par.  in  alt.)  -  log.  sin  (hor.  par.)  -)-  log.  cos.(app.  alt.)  — 10; 

and  add  it,  as   before,   to   the  apparent   altitude  corrected  for 
refraction. 

3.  If  one  of  the  planets  is  the  body  observed,  the  following 
formula  will  serve  for  the  determination  of  the  parallax  in  altitude 
when  the  horizontal  parallax  is  known : 

log.  (par.  in  alt.)  =  log.  (hor.  par.)  +  log.  (cos  appar.  alt.)  —  10. 

Note  1.  The  equatorial  horizontal  parallax  of  the  moon  at  any 
given  time  may  be  obtained  from  the  tables  appended  to  the  work. 
(See  Problem  XIV).  But  it  can  be  had  much  more  readily  from 
the  Nautical  Almanac.  The  equatorial  horizontal  parallax  being 
known,  the  horizontal  parallax  at  any  given  latitude  may  be  ob- 
tained by  subtracting  the  Reduction  of  Parallax,  to  be  found  in 
Table  LXIV.  The  horizontal  parallax  of  any  planet,  the  altitude 
of  which  is  measured,  may  also  be  derived  from  the  Nautical 
Almanac. 

Note  2.  The  fixed  stars  have  no  sensible  parallax,  and  thus  the 
observed  altitude  of  a  star,  corrected  for  refraction,  will  be  its  true 
altitude  at  the  centre  of  the  earth  as  well  as  at  the  station  of  the 
observer. 

Note  3.  If  the  true  altitude  of  a  heavenly  body  is  given,  and 
it  is  required  to  find  the  apparent,  the  rules  for  finding  the  par- 
allax in  altitude  and  the  refraction  are  the  same  as  when  the 


270  ASTRONOMY. 

apparent  altitude  is  given ;  the  tr\ie  altitude  being  used  in 
place  of  the  apparent.  Eut  these  corrections  are  to  be  applied 
with  the  opposite  signs  from  those  used  in  the  determination  of 
the  true  altitude  from  the  apparent;  that  is,  the  parallax  is  to 
be  subtracted,  and  the  refraction  added.  It  will  also  be  more  ac- 
curate to  make  use  of  equa.  (12),  p.  44,  in  the  case  of  the  moon. 
Exam.  1.  The  observed  altitude  of  the  sun  on  the  1st  of  May, 
1837,  being  25°  40'  20",  what  is  its  true  altitude  ? 

Obs.  alt. 26°  40'  20" 

Refraction ■ — 1    56 


True  alt.  at  the  station,  -        -        -  26     38    24 
Parallax  in  alt.  (Table  X)      -        -  +8 


True  altitude        -        -        -        -    26     38    32 
2.  Let  the  apparent  altitude  of  the  moon  at  New  York  on  the 
17th  of  March,  1837,  8h.  P.  M.,  be  66°  10'  44" ;  the  barometer 
30.4  in.  and  the  thermometer  62°  :  required  the  true  altitude. 
Appar.  alt.      -        -      66°  10'  44" 


Mean  refrac.  -  0  25.7 

Corr.  for  +  0.4  in.  bar.  +  0.3 

Corr.  for  +  12°  ther.  —  0.6 


True  refrac.  -        -  0  25.4 


logarithms. 

True  alt.  at  N.  York,  66     10  18.6        cos?  9.60637 
Equa.  par.  by  N.  Almanac,  54'  13" 
Reduc.  for  lat.  40°,  4 

Hor.  par.  at  New  York,     54     9        -        -  sin.  8.19731 


Par.  in  alt.      -        -  21  52  sin.  7.80368 


True  altitude  -  66  32  11 
3.  On  the  ISth  of  February,  1837,  the  true  meridian  altitude 
of  the  planet  Jupiter  at  Greenwich  was  56°  54'  57",  what  was 
its  apparent  altitude  at  the  time  of  the  meridian  passage,  the 
horizontal  parallax  being  taken  at  1".9  as  given  by  the  Nautical 
Almanac  ? 


PROB  IX.    TO  FIND  SUN'S  LONGITUDE,  &C.  FROM  TABLES.      271 

True  alt.        -        -     56"  54'  57"      -    cos.  9.7371 
Hor.  par.  1".9 log.  0.2787 

Par.  in  alt.         -        -        .   _  1.0    -     log.  0.0158 
Refraction         -        -        -   +21.0 


Appar.  alt.      -        -     56     54   37 

4.  What  will  be  the  true  altitude  of  the  sun  on  the  22d  of 
September,  1840,  at  the  time  its  apparent  altitude  is  39°  17'  50"  ? 

Ans.  39°  16'  46". 

5.  Given  29°  33'  30"  the  apparent  altitude  of  the  moon  at 
Philadelphia  on  the  15th  of  June,  1837,  at  9h.  30m.  P.  M.,  and 
58'  33"  the  equatorial  parallax  of  the  moon  at  the  same  time,  to 
find  the  true  altitude.  Ans.  30°  22'  41". 

6.  Given  15°  24'  23"  the  true  altitude  of  Venus,  and  8"  its 
horizontal  parallax,  to  find  the  apparent  altitude. 

Ans.  15°  27'  41". 


PROBLEM  IX, 

To  find  the  Suti's  Longitude.  Semi-diaw,eter,  and  Hourly 
Motion,  for  a  given  time,  from  the  Tables. 

For  the  Longitude. 

When  the  gjiven  time  is  not  for  the  meridian  of  Greenwich, 
reduce  it  to  that  meridian  by  Problem  V  :  and  when  it  is  appa- 
rent time,  convert  it  into  mean  time  by  the  last  problem. 

With  the  mean  time  at  Greenwich,  take  from  Tables  XVIII, 
XIX,  XX,  and  XXI,  the  quantities  corresponding  to  the  year, 
month,  day,  hour,  minute,  and  second  (omittinsf  those  in  the 
last  two  columns),  and  place  them  in  separate  columns  headed 
as  in  Table  XVIII,  and  take  their  sums.*  The  sura  in  the 
column  entitled  M.  Lonsc.  will  be  the  tabular  mean  lonijitude  of 


*  In  adding  quantities  that  are  expressed  in  signs,  degrees,  &c.  reject  12  or  24 
signs  whenever  the  sum  exceeds  either  of  these  quantities.  In  adding  arguments 
expressed  in  100  or  1000,  &c.  parts  of  the  circle,  when  they  consist  of  two  figuroB, 
reject  the  hundreds  from  the  sum  ;  when  of  three  figures,  the  thousands  ;  and  when 
of  four  figures,  the  ten  thousands. 


272  ASTRONOMY. 

the  sun  ;  the  sum  in  the  column  entitled  Long.  Perigee  will  be 
the  tabular  longitude  of  the  sun's  perigee  ;  and  the  sums  in  the 
columns  headed  I,  II,  III,  IV,  V,  N,  will  be  the  arguments  for 
the  small  equations  of  the  sun's  longitude,  and  for  the  equation 
of  the  equinoxes,  which  forms  one  of  them. 

Subtract  the  longitude  of  the  perigee  from  the  sun's  mean 
longitude,  adding  12  signs  when  necessary  to  render  the  sub- 
traction possible  ;  the  remainder  will  be  the  sun's  mean  ano- 
maly. With  the  mean  anomaly  take  the  equation  of  the  sun's 
centre  from  Table  XXV,  and  correct  it  by  estimation,  for  the 
proportional  part  of  the  secular  variation  in  the  interval  between 
the  given  year  and  1840 ;  also,  with  the  arguments  I,  II.  Ill, 
IV,  V,  take  the  corresponding  equations  from  Tables  XXVIII, 
XXX,  XXXI  and  XXXII.  The  equation  of  the  centre  and  the 
four  other  equations,  added  to  the  mean  longitude,  will  give  the 
sun's  True  Longitude,  reckoned  from  the  Mean  Equinox. 

With  the  argnment  N  take  the  equation  of  the  equinoxes  or 
Lunar  Nutation  in  longitude  from  Table  XXVII.  Also  take 
the  Solar  Nutation  in  longitude,  answering  to  the  given  date, 
from  the  same  table.  Apply  these  equations  according  to  their 
signs  to  the  true  longitude  from  the  mean  equinox,  already  found, 
and  add  the  constant  3",  the  result  will  be  the  True  Longitude 
from  the  Apparent  Equinox. 

For  the  Semi-diameter  and  Hourly  Motion. 

With  the  sun's  mean  anomaly,  take  the  Hourly  Motion  and 
Semi-diameter  from  Tables  XXIII  and  XXIV. 

Note  1.  If  the  tenths  of  seconds  be  omitted  in  taking  the  equa- 
tions from  the  tables  of  double  entry,  the  error  cannot  exceed  2" ; 
in  case  the  precaution  is  taken  to  add  a  unit,  whenever  the 
tenths  exceed  .5. 

Note  2.  The  longitude  of  the  sun,  obtained  by  the  foregoing 
rule,  may  differ  about  3"  from  the  same  as  derived  from  the  most 
accurate  solar  tables  now  in  use.  When  there  is  occasion  for 
greater  precision,  take  from  Table  XVIII,  XIX,  and  XX,  the 
quantities  in  the  columns  entitled  VI  and  VII,  along  with  those  in 
the  other  columns.  With  the  sums  in  these  columns,  and  those  in 
the  columns  I,  II,  as  arguments,  take  the  corresponding  equations 
from  Tables  XXIX  and  XXXIII.    Also  with  the  sun's  maan  ano- 


PROB.  IX.   TO  FIND  THE  SUN's  LONGITUDE,  &C. 


273 


maly  take  the  equation  for  the  variable  part  of  the  aberration  from 
Table  XXXIV.  Add  these  three  equations  along  with  the  others 
to  the  mean  longitude,  and  omit  the  addition  of  the  constant  3". 
The  result  will  be  exact  to  within  a  fraction  of  a  second. 

Exam.  1.  Required  the  sun's  longitude,  hourly  motion,  and 
semi-diameter,  on  the  25th  October,  1837,  at  Uh.  27m.  38s.  A.  M. 
mean  time  at  New  York. 

Mean  time  at  N.  York,  Oct.  1837,    24'i-  23^-  27'"-  38^- 
Diff.  of  Long.         -        -        -  4     56      4 


Mean  time  at  Greenwich 

25      4     23     42 

1837    .    .    . 
October    .     . 
25  d.     .     .     . 
4h.     .     .     . 
23m.    .     .     . 
42  s.     .     .     . 

Eq.  Sun's  Cent. 

I.     .     . 
II.  III.     .     . 
II.   IV.     .     . 
II.     V.     .     . 
Const. . 

Lunar  Nutation 
Solar  Nutation 

M.  Long. 

Long.  Perigee. 

I. 

II. 

III. 

IV. 

V. 

N. 

,       O        1         II 

9  10  55  47.2 

8  29     4  54.1 

23  39  19.9 

9  51.4 

56.7 

1.7 

s      O          /         // 

9  10     8     5 
46 

4 

816 

250 

81(1 

6 

280 

748 

66 

0 

.549 

215 

107 

1 

321 

397 

35 

348 

63 

5 

416 

895 

40 

4 

939 

9  10     8  55 
7     3  50  51 

882 

94 

872 

753 

7     3  50  51.0 
11  28  12  43.5 
2.5 
9.0 
7.7 
19.3 
3.0 

1 

9  23  41  56  Mean  Anomaly. 
Sun's  Hourly  Motion,  ....     2'  29".7 
Sun's  Semi-diameter,    .     .     .     .16'  17".2 

7     2     4  16.0 

—  6,1 

—  1.2 

7    2    4    8.7 

2.  Required  the  sun's  longitude,  hourly  motion,  and  semi- 
diameter,  on  the  15th  of  July,  1837,  at  8h.  20m.  40s.  P.  M.  mean 
time  at  Greenwich. 


35 


274 


ASTRONOMY. 


1837     .    .     . 

M.  Long. 

Long.  Peri. 

I. 

IL 

III 

IV. 

V. 

N. 

VI, 

VII. 

,        O          1          II 

9  10  '55  47.2 

,       O         1          II 

9  10     8     5 

816 

280  549 

321 

1 
3481895 

787 

600 

July      .     .     . 

5  '28  24     7.8 

31 

129 

4961,806 

263    41|  27 

569 

17 

15  d.      ... 

13  47  5G.6 

2|473 

38 

C2 

20      3 

2 

508 

2 

8h.     .     .     . 

19  42.8 

11 

1 

1 

11 

30  m.    .     .     . 
40  s.      ... 

49.3 
1.6 

1 

9  10     8  38 

429 

815 

418 

604  392  924 

876 

619  1 

3  23  28  25 

3  23  28  25.3 
11  29  33  10.1 

Eq.  Sun's  Cent. 

6  13  19  47  Mean  Anomaly. 

I.     .     . 

10.7 

II.    III.     .     . 

6.6 

Sun's  Hourly  Motion 2'  23'.1 

II.    IV.     .     . 

5.0 

II.      V.     .     . 

7.7 

Sun's  Semi-diameter,    .     .    .     .15'  45".4 

I.    VI.     .     . 

1.8 

II.  VII.     .     . 

0.2 

Aber.    .     .     . 

0.6 

3  23    2    8.0 

Lunar  Nutation 

—  7.8 

Solar  Nutation 
Sun's  true  long. 

+  0.8 

3  23    2     1.0 

1 

3.  Required  the  sun's  longitude,  hourly  motion,  and  semi- 
diameter,  on  the  10th  of  June,  1838,  at  9h.  45m.  2()s.  A.  M. 
mean  time  at  Philadelphia,  (omitting  the  three  smallest  equations 
of  longitude). 

Ans.  Sun's  longitude,  2^- 19°  11'  57" ;  hourly  motion,  2'  23".3 ; 
semi-diameter,  15'  46". 1. 

4.  Required  the  sun's  longitude,  hourly  motion,  and  semi- 
diameter,  on  the  1st  of  February,  1837,  at  12h.  30m.  15s.  P.  M. 
mean  time  at  Greenwich. 

Ans.  Sun's  longitude,  10^-  13°  1'  44".6 ;  hourly  motion,  2' 
32".  1  ;  semi-diameter,  16'  14".7. 


PROBLEM   X. 


To  find  the  Apparent  Obliquity  of  the  Ecliptic,  for  a  given  time, 
from  the  Tables. 

Take  the  mean  obliquity  for  the  given  year  from  Table  XXII. 
Then  with  the  argument  N,  found  as  in  the  foregoing  problem, 
and  the  given  date,  take  from  Table  XXVII  the  lunar  and  solar 
nutations  of  obliquity ;  apply  these  according  to  their  signs  to 


PROS.  XI.    TO  COMPUTE  SUN's  RIGHT  ASCEN.  AND  DEC.     275 

the  mean  obliquity ;  and  the  result  will  be  the  apparent  ob- 
liquity. 

Exam.  1.  Required  the  apparent  obliquity  of  the  ecliptic  on 
the  15th  of  March,  1839. 
N. 
1839,  -        3 
March,         9 

15d.     -        2 

M.  Obliquity,     23°  27'  36".9 

14 +  9  .1 

Solar  Nutation  for  March  15th,     -        -    +  0  .5 


Apparent  Obliquity,      -         •        -    23   27  46  .5 
2.  Required  the  apparent  obliquity  of  the  ecliptic  on  the  12th 
of  July,  1845.  Ans.  23°  27'  28".0. 


PROBLEM    XI. 

Given  the  Sufi's  Longitude  and  the  Ohliquity  of  the  Ecliptic, 
to  find  his  Right  Ascension  and  Declination* 

Let  u  =  obliquity  of  the  ecliptic  ;  L  =  sun's  longitude ;  R  = 
sun's  right  ascension  ;  and  D  =  sun's  declination  ;  then  to  find  R 
and  D,  we  have, 

log.  tang  R  =  log.  tang  L  +  log.  cos  w  —  10, 
log.  sin  D  =  log.  sin  L  +  log.  sin  w  —  10. 

The  right  ascension  must  always  be  taken  in  the  same  quad- 
rant as  the  longitude.  The  declination  must  be  taken  less  than 
90°  ;  and  it  will  be  north  or  south  according  as  its  trigonometri- 
cal sine  comes  out  positive  or  negative.  » 

Note.  The  sun's  right  ascension  and  declination  are  given  in 
the  Nautical  Almanac  for  each  day  in  the  year  at  noon  on  the 
meridian  of  Greenwich,  and  may  be  found  at  any  intermediate 
time  by  a  proportion. 

Exam.  1.  Given  the  sun's  longitude  205°  23'  50",  and  the  ob- 


»  The  obliquity  of  the  ecliptic  at  any  given  time  for  which  the  sun's  longitude 
is  known,  is  found  by  the  foregoing  Problem. 


276  ASTRONOMY. 

liquity  of  the  ecliptic  23°  27'  36",  to  find  his  right  ascension  tind 

declination. 

L  =  205°  23'    50"       -        -        -        tan.    9.67649 
u  =    23    27     36        -        -        -        cos.    9.96253 


R  =  203     32       5        -        -        -        tan.    9.63902 


L  =  205     23     50        -        -        -        sin.    9.63235  — 
6J  =    23     27     36        -        -        -        sin.    9.60000 


D=      9    49     52S.    -        -        -        sin.    9.23235  — 
2,  The  obliquity  of  the  ecliptic  being  23°  27'  30",  required 

the  sun's  right  ascension  and  declination  when  his  longitude  is 

44°  18'  25". 
Ans.  Rio-ht  ascension  41°  50'  30",  and  declination  16°  8'  40"  N. 


PROBLEM  XII. 

Given  the  Sun^s  Right  Ascension^  and  the  Obliquity  of  the 
Ecliptic,  to  find  his  Longitude  and  Declination. 
Using  the  same  notation  as  in  the  last  problem,  we  have,  to 
find  the  longitude  and  declination, 

log,  tang  L  —  log.  tang  R  +  ar.  co.  log.  cos  u, 
log.  tang  D  =  log.  sin  R  -|-  log.  tang  w  —  10. 
Exam.  1.  What  is  the  longitude  and  declination  of  the  sun, 
when  his  right  ascension  is  142°  11'  34",  and  the  obliquity  of 
the  ecliptic  23°  27'  40"  ? 

R  =  142°ir34"        -        -        -        tan.  9.88979 — 
w  =    23    27  40  -        -    Ar.  Co.cos.   0.03747 


L  =  139    46   30  «        -        -        tan.  9.92726  — 


R  =  142    11    34  -        -        -        sin.  9.78746 

w  =    23    27   40  -        -        -        tan.  9.63750 


D  =    14    53   55  N      -        -        -        tan.  9.42496 
2.  Given  the  sun's  right  ascension  310°  25'  11,  and  the  obli- 
quity of  the  ecliptic  23°  27'  35",  to  find  the  longitude  and  declina- 
tion. 

Ans.  Longitude  307°  59'  57",  and  declination  18°  17'  0"  S. 


PROS.  XIII.  TO  FIND  THE  SUn's  ANGLE  OP  POSITION.         277 


PROBLEM   XIII. 

The  Sun^s  Lo7igitude  and  the   Obliquity  of  the  Ecliptic 

being  given,  to  find  the  Angle  of  Position. 
liet  p  =  angle  of  position  ;  w  =  obliquity  of  the  ecliptic  ;  and 
L  =  sun's  longitude.     Then, 

log.  tang  p  —  log.  cos  L  +  log.  tang  u  —  10. 

The  angle  of  position  is  always  less  than  90°.  The  northern 
part  of  the  circle  of  latitude  will  be  to  the  xoest  or  east  of  the 
northern  part  of  the  circle  of  declination,  according  as  the  sign 
of  the  tangent  of  the  angle  of  position  is  positive  or  negative. 

Exam.  1.  Given  the  sun's  longitude  24°  15'  20",  and  the  obli- 
quity of  the  ecliptic  23°  27'  32",  required  the  angle  of  position. 
L=  24°  15'  20"      -        -        -        -        cos.    9.95986 
cj=23    27   32        -        -        -        -        tan.    9.63745 


p=  21    35   10        -        -        -        -        tan.    9.59731 
The  northern  part  of  the  circle  of  latitude  lies  to  the  west  of 
the  circle  of  declination. 

2.  When  the  sun's  longitude  is  120°  18'  55",  and  the  obli- 
quity of  the  ecliptic  23°  27'  30",  what  is  the  angle  of  position  ? 
Ans.  12°  21'  17" ;  and  the  northern  part  of  the  circle  of  lati- 
tude lies  to  the  east  of  the  circle  of  declination. 


PROBLEM   XIV. 

To  find  from  the    Tables,  the  Moon^s  Longitude,   Latitude, 

Equatorial  Parallax,    Semi-diam,eter,   and  Hourly  Motion 

in  Longitude  and  Latitude,  for  a  given  time. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich, 

reduce  it  to  that  meridian,  and  when  it  is  apparent  time  convert 

it  into  mean  time. 

Take  from  Table  XXXV,  and  the  following  tables,  the  argu- 
ments numbered  1,  2,  3,  (fee,  to  20,  for  the  given  year,  and  their 
variations  for  the  given  month,  days,  &c.,  and  find  the  sums  of 
the  numbers  for  the  diiferent  arguments  respectively ;  rejecting 


278  ASTRONOMY. 

the  hundred  thousands  and  also  the  units  in  the  first,  the  ten 
thousands  in  the  next  eight,  and  the  thousands  in  the  others. 

The  resulting  quantities  will  be  the  arguments  for  the  first 
twenty  equations  of  longitude. 

With  the  same  time,  take  from  the  same  tables  the  remaining 
arguments  with  their  variations,  entitled  Evection,  Anomaly,  Va- 
riation, Longitude,  Supplement  of  the  Node,  II,  V,  VI,  VII,  VIII, 
IX,  and  X,  and  add  the  quantities  in  the  column  for  the  Supple- 
ment of  the  Node. 

For  the  Longitude. 

With  the  first  twenty  arguments  of  longitude,  take  from  Tables 
XLI  to  XLVI,  inclusive,  the  corresponding  equations  ;  and  with 
the  Supplement  of  the  Node  for  another  argument,  take  the  cor- 
responding equation  from  Table  XLIX.  Place  these  twenty-one 
equations  in  a  single  column,  headed  Eqs.  of  Long. ;  and  write 
beneath  them  the  constant  55".  Find  the  sum  of  the  whole, 
and  place  it  in  the  column  of  Evection.  Then  the  sum  of  the 
quantities  in  this  column  will  be  the  corrected  argument  of 
Evection. 

With  the  corrected  argument  of  Evection,  take  the  Evection 
from  Table  L,  and  add  it  to  the  sum  in  the  column  of  Eqs.  of 
Long.  Place  this  in  the  column  of  Anomaly.  Then  the  sum  of 
the  quantities  in  this  column  will  be  the  corrected  Anomaly. 

With  the  corrected  Anomaly,  take  the  Equation  of  the  Centre 
from  Table  LI,  and  add  it  to  the  last  sum  in  the  column  of 
Eqs.  of  Long.  Place  the  resulting  sum  in  the  column  of  Va- 
riation. Then  the  sum  of  the  quantities  in  this  column  will  be 
the  corrected  argument  of  Variation. 

With  the  corrected  argument  of  Variation,  take  the  variation 
from  Table  LII,  and  add  it  to  the  last  sum  in  the  column  of 
Eqs.  of  Long. ;  the  result  will  be  the  sum  of  the  principal 
equations  of  the  Orbit  Longitude,  amounting  in  all  to  twenty- 
four,  and  the  constants  subtracted  for  the  other  equations.  Place 
this  sum  in  the  column  of  Longitude.  Then,  the  sum  of  the 
quantities  in  this  column  will  be  the  Orbit  Longitude  of  the 
Moon,  reckoned  from  the  mean  equinox. 

Add  the  orbit  longitude  to  the  supplement  of  the  node,  and 
the  resulting  sum  will  be  the  argument  of  Reduction. 


PROB.    XIV.    TO    FIND    THE    MOOn's    LONGITUDE,  &C.         279 

With  the  arg-iiment  of  Reduction,  take  the  Reduction  from 
Table  LIII,  and  add  it  to  the  Orbit  Longitude.  The  sum  will 
be  the  Longitude  as  reckoned  from  the  mean  equinox.  With 
the  Supplement  of  the  Node,  take  the  Nutation  in  Longitude 
from  Table  XXVII,  and  apply  it,  according  to  its  sign,  to  the 
longitude  from  the  mean  equinox.  The  result  will  be  the 
Moon's  True  Longitude  from  the  Apparent  Equinox. 

For  the  Latitude. 

The  argument  of  the  Reduction  is  also  the  1st  argument  of 
Latitude.  Place  the  sum  of  the  first  twenty-four  equations  of 
Longitude,  taken  to  the  nearest  minute,  in  the  column  of  Arg. 
II.  Find  the  sum  of  the  quantities  in  this  column,  and  it  will 
be  the  Arg.  II  of  Latitude,  corrected.  The  Moon's  true  Longi- 
tude is  the  3d  argument  of  Latitude.  The  20th  argument 
of  Lonaitude  is  the  4th  argument  of  Latitude.  Take  from 
Table  LVIII  the  thousandth  parts  of  the  circle,  answering  to 
the  degrees  and  minutes  in  the  sum  of  the  first  twenty-four 
equations  of  longitude  ;  and  place  it  in  the  columns  V,  VI,  VII, 
VIII,  and  IX  ;  but  not  in  the  column  X.  Then  the  sums  of 
the  quantities  in  columns  V,  VI,  VII,  VIII,  IX  and  X,  rejecting 
the  thousands,  will  be  the  5th,  6th,  7th,  8th,  9th,  and  10th  ar- 
guments of  Latitude. 

With  the  Arg.  I  of  Latitude,  take  the  moon's  distance  from 
the  North  Pole  of  the  Ecliptic,  from  Table  LV ;  and  with  the 
remaining  nine  arguments  of  latitude,  take  the  corresponding 
equations  from  Tables  LVI,  LVII  and  LIX.  The  sum  of  these 
quantities  increased  by  8",  Avill  be  the  Moon's  true  distance 
from  the  North  Pole  of  the  Ecliptic.  The  difference  between 
this  distance  and  90°  will  be  the  Moon's  true  latitude  ;  which 
will  be  North  or  South,  according  as  the  distance  is  less  or 
greater  than  90°. 

For  the  Equatorial  Parallax. 
With  the  corrected  arguments,  Erection,  Anomaly,  and  Vari- 
ation, take  out  the  corresponding  quantities  from  Tables  LXI, 
LXII,  and  LXIII.     Their  sum  increased  by  7",  will  be  the  Equa- 
torial Parallax. 


280  ASTRONOMY. 

For  the  Semi-diameter. 
With  the  Equatorial  Parallax  as  an  argument,  take  out  the 
moon's  semi-diameter  from  Table  LXV. 

For  the  Hourly  Motion  in  Longitude. 

With  the  arguments  2,  3,  4,  5,  and  6  of  Longitude,  rejecting 
the  two  right-hand  figures  in  each,  take  the  corresponding 
equations  of  the  hourly  motion  in  longitude  from  Table  LXVII. 
Find  the  sum  of  these  equations,  and  the  constant  3",  eind  with 
this  sum  at  the  top,  and  the  corrected  argument  of  the  Evection 
at  the  side,  take  the  corresponding  equation  from  Table  LXIX  ; 
also  with  the  corrected  argument  of  the  Evection,  take  the  cor- 
responding equation  from  Table  LXVIII. 

Add  these  equations  to  the  sum  just  found,  and  with  the  re- 
sulting sum  at  the  top,  and  the  correct  anomaly  at  the  side,  take 
the  corresponding  equation  from  Table  LXX ;  also  with  the 
corrected  anomaly,  take  the  corresponding  equation  from  Table 
LXXI. 

Add  these  two  equations  to  the  sum  last  found,  and  with  the 
resulting  sum  at  the  top,  and  the  corrected  argument  of  the 
Variation  at  the  side,  take  the  corresponding  equation  from 
Table  LXXII.  With  the  corrected  argument  of  the  Variation, 
take  the  corresponding  equation  from  Table  LXXIII. 

Add  these  two  equations  to  the  sum  last  found,  and  with  the 
resulting  sum  at  the  top,  and  the  argument  of  the  Reduction  at 
the  side,  take  the  corresponding  equation  from  Table  LXXIV. 
Also,  with  the  argument  of  the  Reduction  take  the  correspond- 
ing equation  from  Table  LXXV.  These  two  equations,  added 
to  the  last  sum,  will  give  the  sum  of  the  principal  equations  of 
the  hourly  motion  in  longitude,  and  the  constants  subtracted 
for  the  others.  To  this  add  the  constant  27'  24".0,  and  the  re- 
sult will  be  the  Moon's  Hourly  Motion  in  Longitude. 

For  the  Hourly  Motion  in  Latitude. 
With  the  Argument  I  of  Latitude,  take  the  corresponding 
equation  from  Table  LXXIX.  With  this  equation,  and  the 
sum  of  all  the  equations  of  the  hourly  motion  in  longitude, 
except  the  last  two,  take  the  corresponding  equation  from 
Table  LXXXL    With  the  Argument  II  of  Latitude,  take  the 


PROB.    XIV.    TO    FIND    THE    MOON's    LONGITUDE,  fcC.         281 

corresponding  equation  from  Table  LXXXII.  And  with  this 
equation  at  the  top,  and  the  sum  of  all  the  equations  of  the 
hourly  motion  in  longitude,  except  the  last  two,  take  the  equa- 
tion from  Table  LXXXIII.  Find  the  sum  of  these  four  equa- 
tions and  the  constant  1".  To  the  resulting  sum  apply  the 
constant  — 2'  37".2.  The  difference  will  be  the  Moon's  true 
Hourly  Motion  in  Latitude.  The  moon  will  be  tending  North 
or  South,  according  as  the  sign  is  positive  or  negative. 

Note.  The  errors  of  the  results  obtained  by  the  foregoing  rules, 
occasioned  by  the  neglect  of  the  smaller  equations,  cannot  exceed 
for  the  longitude  15",  for  the  latitude  8",  for  the  parallax  7",  for 
the  hourly  motion  in  longitude  5",  and  for  the  hourly  motion  in 
latitude  3" ;  and  they  will  generally  be  very  much  less.  When 
greater  accuracy  is  required,  take  from  Tables  XXXV  to  XXXIX 
the  arguments  from  21  to  31,  along  with  those  from  1  to  20,  and 
their  variations.  The  sums  of  the  numbers  for  the  different  argu- 
ments, respectively,  will  be  the  arguments  of  eleven  small  addi- 
tional equations  of  longitude.  Also  take  from  the  same  tables 
the  arguments  entitled  XI  and  XII,  along  with  those  in  the  pre- 
ceding columns.  Retain  the  right-hand  figure  of  the  sum  in 
column  1  of  arguments,  and  conceive  a  cypher  to  be  annexed  to 
each  number  in  the  columns  of  arguments  of  Table  XLI.  The 
numbers  in  the  columns  entitled  Diff.for  10,  will  then  be  the 
differences  for  a  variation  of  100  in  the  argument. 

For  the  Longitude.  With  the  arguments  21  to  31,  take  the  cor- 
responding equations  from  Tables  XLVII  and  XLVIII,  and  place 
them  in  the  same  column  with  the  equations  taken  out  with  the  ar- 
guments 1, 2,  &.C.  to  20.  Take  alsoequation  32  from  Table  XLIX, 
as  before.  Find  the  sum,of  the  whole,  (omitting  the  constant  55") 
and  then  continue  on  as  above.  The  longitude  from  the  mean  equi- 
nox being  found,  take  the  lunar  nutation  in  longitude  from  Table 
LIV,  and  the  solar  nutation  answering  to  the  given  date  from 
Table  XXVII.  Apply  them  both,  according  to  their  sign,  to  the 
longitude  from  the  mean  equinox,  and  the  result  will  be  the  more 
exact  longitude  from  the  apparent  equinox,  required. 
36 


282  ASTRONOMY. 

For  the  Latitude.  With  tlie  arguments  XI  and  XII,  take  the 
corresponding  equations  from  Table  LIX.  Add  these  with  the 
other  equations,  and  omit  the  constant  8".  The  difference  between 
the  sum  and  90°  will  be  the  more  exact  latitude. 

For  the  Equatorial  Parallax.  With  the  arguments  1.  2,  4, 
5,  6,  8,  9,  12,  13,  take  the  corresponding  equations  from  Table 
LX.  Find  the  sum  of  these  and  the  other  equations,  omitting 
the  constant  7",  and  it  will  be  the  more  exact  value  of  the 
Parallax. 

For  the  Hourly  Motion  in  Longitude.  With  the  arguments 
1,  7,  8,  9, 10,  11,  12,  13,  14, 15, 16,  17,  and  18,  of  longitude,  along 
with  the  arguments  2,  3,  4,  5,  and  6,  heretofore  used,  take  the 
corresponding  equations  from  Table  LXVII.  Find  the  sum  of 
the  whole,  omitting  the  constant  3",  and  proceed  as  in  the  rule 
already  given. 

To  obtain  the  motion  in  longitude  for  the  hour  which  precedes 
or  follows  the  given  time,  with  the  arguments  of  Tables  LXX 
LXXII,  and  LXXIV,  take  the  equation  from  Tables  LXXVI  and 
LXXVIL  Also,  with  the  arguments  of  Evection,  Anomaly,  Va- 
riation, and  Reduction,  take  the  equations  from  Tables  LXXVIII. 
Find  the  sum  of  all  these  equations.  Then,  for  the  hour  which 
follows  the  given  time,  add  this  sum  to  the  hourly  motion  at  the 
given  time  already  found,  and  subtract  2".0  ;  for  the  hour  which 
precedes,  subtract  it  from  the  same  quantity,  and  add  2".0. 

It  will  expedite  the  calculation  to  take  the  equations  of  the 
second  order  from  the  tables,  at  the  same  time  with  those  of  the 
first  order  which  have  the  same  arguments. 

For  the  Hourly  Motion  in  Latitude.  The  moon's  hourly 
motion  in  latitude  may  be  had  more  exactly  by  taking  with  the 
arguments  of  Latitude  V,  VI,  &c.  to  XII,  the  corresponding  equa- 
tions from  Table  LXXX,  and  finding  the  sum  of  these  and  the 
other  equations  of  the  hourly  motion  in  latitude. 

To  obtain  the  moon's  motion  in  latitude  for  the  hour  which 
precedes  or  follows  the  given  time,  with  the  Argument  I  of  Lati- 
tude take  the  equation  from  Table  LXXXIV,  and  with  this  equa- 
tion and  the  sum  of  all  the  equations  of  the  hourly  motion  in  lon- 
gitude except  the  two  last,  take  the  equation  from  Table  LXXXV. 


PROB.  XIV.    TO    FIND    THE    MOOn's    LONGITUDE,    &C.  283 

Find  the  sum  of  these  two  equations.  Then,  for  the  hour  which 
follows  the  given  time,  add  this  sum  to  the  Hourly  Motion  in 
Latitude  already  found,  and  subtract  1".3  ;  and  for  the  hour  which 
precedes,  subtract  it  from  the  same  quantity,  and  add  1".3. 

It  will  also  be  more  exact  to  enter  Table  LXXXI  with  the  sum 
of  all  the  other  equations,  diminished  by  6",  instead  of  the  last 
equation,  for  the  argument  at  the  side.  The  numbers  over  the 
tops  of  the  columns  in  Table  LXXXI  are  the  common  differences 
of  the  consecutive  numbers  in  the  columns.  The  numbers  in  the 
last  column  are  the  common  differences  of  the  consecutive  num- 
bers in  the  same  horizontal  line. 

Exam.  L  Required  the  moon's  longitude,  latitude,  equatorial 
parallax,  semi-diameter,  and  hourly  motions  in  longitude  and  lat- 
itude, on  the  14th  of  October,  1838,  at  6h.  54m.  34s.  P.  M.  mean 
time  at  New  York. 

Mean  time  at  New  York,  October,    14''-    6'^-  54'"-  34=^- 
Diff.  of  Long.     ...        -  4     56      4 


Mean  time  at  Greenwich,  October,  14     11     50     38 


284 


ASTRONOMY. 


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PROB.  XIV.    TO  FIND  THE  MOON's  LONGITUDE,  &C. 


285 


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PROB.  XIV.    TO  FIND  THE  MOON's  LONGITUDE,  &C. 


287 


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288  ASTRONOMY. 

Exam.  2.  Required  the  moon's  longitude,  latitude,  equatorial 
parallax,  semi-diameter,  and  hourly  motions  in  longitude  and 
latitude,  on  the  9th  of  April,  1838,  at  8h.  58m.  19sec.  P.  M.  mean 
time  at  Washington. 

Ans.  Long.  6^-  19°  45'  31".2  ;  lat.  36'  21".9  S. ;  equat.  par. 
54'  36".3;  semi-diameter  14'  52".7  ;  hor.  mot.  in  long.  30' 
15".2  ;  and  hor.  mot.  in  lat.  2'  47".0,  tending  south.* 


PROBLEM    XV. 

The  Moon^s  Equatorial  Parallax,  and  the  Latlttide  of  a  Place, 
being  given,  to  find  the  Reduced  Parallax  a7id  Latitude. 
With  the  latitude  of  the  place,  take  the  reductions  from  Table 
LXIV,  and  subtract  them  from  the  Parallax  and  Latitude. 

Exam.  1.  Given  the  equatorial  parallax  55'  15",  and  the  lati- 
tude of  New  York  40^  42'  49"  N.,  to  find  the  reduced  parallax 
and  latitude. 

Equatorial  parallax       -         -         -         -     55'  15" 
Reduction 5 


Reduced  parallax  -        -        -        -     55  10 


Latitude  of  New  York  -        -        -      40°  42'  40"  N. 
Reduction 11  20 


Reduced  Lat.  of  New  York  -        -      40     31  20 

2.  Given  the  equatorial  parallax  60'  36"  and  the  latitude  of 
Baltimore  39°  17'  13"  N.,  to  find  the  reduced  parallax  and 
latitude. 

Ans.  Reduced  par.  60'  32",  and  reduced  lat.  39°  5'  59". 

3.  Given  the  equatorial  parallax  57'  22",  and  the  latitude  of 
New  Orleans  29°  57'  45"  N.,  to  find  the  reduced  parallax  and 
latitude. 

Ans.  Reduced  par.  57'  19",  and  reduced  lat.  29°  47'  50". 


•  The  smaller  equations  were  omitted  in  working  this  example. 


PROR.    XVX.    TO  FTNn  THF.   T.ONO.    AND   ALT.  OF  THE  NONAG.        289 


PROBLEM    XVI. 

To  find  the  Longitude  and  Altitude  of  the  Nonagesimal  Degree 
of  the  Ecliptic,  for  a  given  time  and  place. 

For  the  given  time  reduced  to  mean  time  at  Greenwich,  find 
the  sun's  mean  longitude  and  the  argument  N  from  Tables 
XVIII,  XIX,  XX,  and  XXI.  To  the  sun's  mean  longitude, 
apply  accordino-  to  its  sign  the  nutation  in  right  ascension,  taken 
from  Table  XXVII  with  argument  N  ;  and  the  result  will  be 
the  sun's  mean  longitude,  reckoned  from  the  true  equinox. 

Reduce  the  mean  time  of  day  at  the  given  place,  expressed 
astronomically,  to  degrees,  d:c.,  and  add  it  to  the  sun's  mean  lon- 
gitude from  the  true  equinox.  The  sum,  rejecting  360°,  when 
it  exceeds  that  quantity,  will  be  the  right  ascension  of  the  mid- 
heaven,  or  the  sidereal  time  in  degrees. 

Next,  find  the  reduced  latitude  of  the  place  by  Problem  XV ; 
and  when  it  is  7iorth,  subtract  it  from  90°  ;  but  when  it  is  south, 
add  it  to  90°  ;  the  sum  or  difference  will  be  the  reduced  distance 
of  the  place  frotn  the  north  pole. 

Also  take  the  obliquity  of  the  ecliptic  for  the  given  year  from 
Table  XXII.* 

These  three  quantities  having  been  found,  the  longitude  and 
altitude  of  the  nonagesimal  degree  may  be  computed  from  the 
following  formulae  : 

log.  COS.  ^  (H  —  u)  —  log.  COS.  i-  (H  +  w)  =  A  .  .  .  (1) ; 
log.  tang  ^  (H  —  c^)  +  10  —  log.  tang^  (H  +  w)  -  B  .  .  .  (2) ; 
log.  tang    E  =  A  +  log.  tang  ^{S  —  90°)  .  .  .  (3) ; 
log.  tang    F  =  log.  tang  E  +  B  .  .  .  (4) ; 
N  -  E  +  F  +  90°  .  .  .  (5) ; 

log.  tang ^h  =  log.  cos.  E  +  log.  tang ^  (H  +  w)  +  ar.  co.  log. 
COS.  F  —  20  .  .  .  ("g)  ; 


•  If  great  precision  is  required,  the  apparent  obliquity  is  to  be  used  in  place  of 
the  mean.     (See  Prob.  X.) 

37 


290  ASTROXOMV. 

in  which, 

H  =  the  reduced  distance  of  the  place  from  the  north  pole  ; 

w   =  the  Obliquity  of  the  Ecliptic  ; 

S  =  the  Sidereal  Time  converted  into  degrees  ; 

N  =  the  required  Longitude  of  the  Nonagesimal  ; 

h   =  the  required  Altitude  of  the  Nonagesimal  ; 

E  and  F  are  auxiliary  angles. 
We  first  find  the  logarithmic  sums  A  and  B.     With  these  we 
determine  the  angles  E  and  F  by  formulae  (3)  and  (4),  and  with 
these  again  N  and  h  by  formulae  (5)  and  (6). 

The  angles  E,  F  are  to  be  taken  less  than  180°  ;  and  less  or 
greater  than  90°,  according  as  the  sign  of  their  tangent  proves 
to  be  positive  or  negative. 

Note  1.  In  case  the  given  place  lies  within  the  arctic  circle, 
we  must  take,  in  place  of  formula  (5),  the  following  : 

N  =  E  —  F  +  90°. 

« 

Note  2.  As  the  obliquity  of  the  ecliptic  varies  but  slowly  from 
year  to  year,  the  values  which  have  once  been  found  for  the 
logarithms  A,  B,  and  C,  will  answer  for  several  years  from  the 
date  of  their  determination,  unless  very  great  accuracy  is 
required. 

Note  3.  The  angle  h  derived  from  formula  (6),  is  the  distance 
of  the  zenith  of  the  given  place  from  the  north  pole  of  the  eclip- 
tic. This  is  not  always  equal  to  the  altitude  of  the  nonagesimal. 
Throughout  the  southern  hemisphere,  and  frequently  in  the 
northern  near  the  equator,  it  is  the  supplement  of  the  altitude. 
In  employing  this  angle  in  the  following  Problem,  it  is,  however, 
for  the  sake  of  simplicity,  called  the  altitude  of  the  nonagesimal 
in  all  cases. 

Exam.  1.  Required  the  longitude  and  altitude  of  the  nonagesi- 
mal degree  of  the  ecliptic  at  New  York,  on  the  18th  of  Septem- 
ber, 1838,  at  3h.  52m.  56sec.  P.  M.  mean  time. 

The  sun's  mean  longitude  taken  from  the  tables,  for  the  given 
time,  is  5^-  27°  19'  17",  and  the  argument  N  is  987.  The  nuta- 
tion taken  from  Table  XXVIl  with  argument  N  is  —  1". 
Hence,  the  sun's  mean  longitude  from  the  true  equinox  is  5'-  27° 


PROB.   XVI.    TO  FIND  THE  LONG.   AND  ALT.   OF  THE  NONAG.    291 


19'  16".     The  given  time  of  day,  expressed  astronomically,  is 
3h.  52m.  56sec.  ;  which  in  degrees  is  58°  14'  0". 

The  reduced  latitude  of  New  York,  found  by  Problem  XV,  is 
40°  31'  20",  and  this  taken  from  90°  leaves  the  polar  distance  49° 
28'  40".  The  obliquity  of  the  ecliptic,  derived  from  Table 
XXII,  is  23°  27'  37". 

Given  time  in  degrees  - 

Sun's  mean  longitude    - 

Sidereal  time  in  degrees  (S)   - 


-  58° 

-  177 

14'  0' 
19  16 

-  235 
90 

33  16 

2)  145 

33  16 

H  - 

u      - 

-  49° 

-  23 

28'  40" 
27  37 

.i(S  — 90) 

•  COS.  9.98870  - 
-  COS.  9.90535  - 

72  46  38 

Diff. 
Sum 

-  26 

-  72 

1  3 

56  17 

^diff. 
^  sum 

-  13 

-  36 

0  31  - 

28  8  ■ 

tan. +  10,19.36366 
tan.  C.  9.86871 

^(S  — 90°)72    46  38 
E    -        -     75     38  55 


A.     0.08335 

0.50866 


tan.  0.59201 
B.     9.49495 


COS. 


B.  9.49495 

9.39422 

C.  9.86871 


50     41  55 
90      0     0 


tan.   0.08696    -  Ar.co.  cos.  0.19832 


|alt..non.  16°    7' 54"    tan.  9.46125 

long.  non.  216     20  50  

alt.  non.  32    15  48 
2.  Required  the  longitude  and  altitude  of  the  nonagesimal 
degree  of  the  ecliptic  at  New  York,  on  the  10th  of  May,  1838,  at 
llh.  33m.  56sec.  P.  M,  mean  time. 

Ans.  Long.  200°  12'  23",  and  alt.  37°  0'  34". 


292  ASTRONOMY. 


PROBLEM    XVII. 


To  find  the  Apparent  Longitude  and  Latitude,  as  affected  by 
Parallax,  and  the  Augmented  Semi-diameter  of  the  Moon  ;  the 
Moon^s  True  Longitude,  Latitude,  Horizontal  Semi-diameter, 
and  Equatorial  Parallax,  and  the  Longitude  and  Altitude  of 
the  Nonagesimal  Degree  of  the  Ecliptic,  being  given. 

We  have  for  the  resolution  of  this  Problem  the  following 
formulae  : 

log.  X—  log.P+log.  COS.  A-f  ar.co.log.cos.X  — 10. .  (1); 

c  =  log.  X  +  log.  tang  h  —  10  .  .  .  (2) ; 
log.  M  =  c  +  log.  sin  K  —  10  .  .  .  (3) ; 
log.  w'  =  c  +  log.  sin  (K  +  m)  —  10  .  .  .  (4) ; 
log.  p  =c+  log.  sin  (K  +  m')  —  10  .  .  .  (5) ; 
Appar.  long.    =  true  long.  +  ^  .  .  .  (6) ; 

log.  tangX'  =  log.  p  +  ar.  co.  log.  cos.  X  +  ar.  co.  log.  u  + 
log.  sin  (X  —  j:)  —  10  .  .  .  (7)  ; 
log.  V  =  log.  P  +  log.  cos.  h  -\-  log.  COS.  X'  —  10  .  .  .  (8) ; 
log.  z  =  log.  V  -\-  log.  tang  h  +  log.  tang  X'  -\-  log.  cos. 
(K+ip)-30;  .  .  (9); 
*   =  v  —  z  .  .  .  (10) ; 
Appar.  lat.    =  true  lat.  —  if  .  ;  .  (11) ; 
log.  R'  =  log.  p  -\-  ar.  co.  log.  cos.  X  +  ar.  co.  log.  u  -\-  log. 
COS.  X'  +  log.  R—  10  .  .  .  (12); 

in  which, 

P  =  the  Reduced  Parallax  of  the  Moon  ; 

h   =  the  Altitude  of  the  Nonagesimal  ; 

X   =  the  True  Latitude  of  the  Moon  (minus  when  south) ; 

K  =  the  Longitude  of  the  Moon,  minus  the  longitude  of 
the  Nonagesimal ; 

p  =  the  required  Parallax  in  Longitude  ; 

X'  =  the  approximate  Apparent  Latitude  of  the  Moon  ; 

«  =  the  required  Parallax  in  Latitude  ; 


PROB.  XVII.    TO  FIND  THE  MOON's  APPAR.  LONG.  AND  LAT.    293 

R'  =:  the  True  Semi-diameter  of  the  Moon  ; 

R'  =  the  Augmented  Semi-diameter  of  the  Moon  ; 

x^  M,  m',  V,  z^  are  auxiliary  arcs. 

Formulae  (1),  (2),  (3),  (4),  and  (.5),  being  resolved  in  succes- 
sion, we  derive  the  apparent  longitude  from  formula  (6) ;  then  the 
apparent  latitude  from  equations  (7),  (8),  (9),  (10),  (11) ;  and 
lastly,  the  augmented  semi-diameter  from  equation  12. 

The  latitude  of  the  moon  must  be  affected  with  the  negative 
sign  when  south  ;  and  the  apparent  latitude  will  be  south  when 
it  comes  out  negative.  In  performing  the  operations,  it  is  to  be 
remembered  that  the  cosine  of  a  negative  arc  has  the  same  sign 
as  the  cosine  of  a  positive  arc  of  an  equal  number  of  degrees  ; 
but  that  the  sine  or  tangent  of  a  negative  arc  has  the  opposite 
siffn  from  the  sine  or  tangent  of  an  equal  positive  arc.  Attention 
must  also  be  paid  to  the  signs  in  the  addition  and  subtraction  of 
arcs.  Thus,  two  arcs  affected  with  essential  signs,  which  are  to 
be  added  to  each  other,  are  to  bo  added  arithmetically,  when 
they  have  like  signs,  but  subtracted  if  they  have  unlike  signs  ;  and 
when  one  arc  is  to  be  taken  from  another,  its  sign  is  to  be  changed, 
and  the  two  united  according  to  their  signs.  An  arithmetical 
sum,  when  taken,  will  have  the  same  sign  as  each  of  the  arcs ; 
and  an  arithmetical  difference  the  same  sign  as  the  greater 
arc. 

The  use  of  negative  arcs  may  be  avoided,  though  the  calcula- 
tion would  be  somewhat  longer,  by  using  the  true  polar  distance 
c?,  and  the  approximate  apparent  polar  distance  </',  in  place  of  X 
and  X',  substitutinof  sin  d  for  cos.  X,  cos.  {d  +  x)  for  sin  (X  — x), 
sin  d'  for  cos.  X',  log.  co-tang  rf'  for  log.  tang  X' ;  and  .observing 
that  p  is  to  be  subtracted  from  the  true  longitude  in  case  the 
longitude  of  the  nonagesimal  exceeds  the  longitude  of  the  moon  ; 
that  z,  when  it  comes  out  negative,  is  to  be  added  to  v,  which  is 
always  positive  to  the  north  of  the  tropic,  otherwise  subtracted ; 
and  that  the  parallax  in  latitude  is  to  be  applied  according  to  its 
sign  to  the  true  polar  distance. 

In  seeking  for  the  logarithms  of  the  trigonometrical  lines,  it 
will  be  sufficient  to  take  those  answering  to  the  nearest  tens  of 
seconds. 

Note  1.  When  great  accuracy  is  not  desired,  w'  may  be  taken 


294  ASTRONOMY. 

for  py  from  which  it  can  never  differ  more  than  a  fraction  of  a 
second. 

2.  In  solar  eclipses,  the  moon's  latitude  is  very  small,  and  for- 
mula (7)  may  be  changed  into  the  following, 

log.  X'  =  log.  p  +  ar.  CO.  log.  cos.  X  -f  ar.  co.  log.  u  -f  log.  (X  —  .r)  —  10 

and  COS.  X'  omitted  in  formula  (12)  without  material  error. 

Formulae  (8),  (9),  (10),  and  (11),  may  also  now  be  dispensed 
with,  unless  very  great  precision  is  desired,  and  the  value  of  X' 
given  by  the  above  formula  taken  for  the  apparent  latitude. 

It  is  to  be  observed  also,  that  in  eclipses  of  the  sun  P  is  taken 
equal  to  the  reduced  parallax  of  the  moon  minus  the  sun's  hori- 
zontal parallax.  By  this  the  parallax  of  the  sun  in  longitude  and 
latitude  is  referred  to  the  moon,  and  the  relative  apparent  places 
of  the  sun  and  moon  are  correctly  obtained,  without  the  neces- 
sity of  a  separate  computation  of  the  sun's  parallax  in  longitude 
and  latitude. 

Exam.  1.  About  the  time  of  the  middle  of  the  occultation  of 
the  star  Antares,  on  the  10th  of  May,  1838,  the  moon's  longitude, 
by  the  Connaissance  des  Tems,  was  247°  37'  6". 7  ;  latitude  4° 
14'  14".7  S.;  semi-diameter  15'  24".2 ;  and  equatorial  parallax 
56'  3l".7  ;  and  the  longitude  of  the  nonagesimal  at  New  York 
was  200°  12'  23" ;  the  altitude  37°  0'  34"  ;  required  the  appa- 
rent longitude  and  latitude,  and  the  augmented  semi-diameter  of 
the  moon  at  New  York,  at  the  time  in  question. 

Equat.  par.      56'  31".7  Moon's  long.    247°  37'      7" 

Reduction  4  .6  Long,  nonag.    200     12     23 

P=56  27  .1 


P 
h 


.1 

K  = 

47  24  44 

h   = 

37   0  34 

X  = 

-4  14  14.7 

- 

3387".l  - 

- 

log.  3.52983 

37° 

0'  34"  - 

- 

cos.  9.90230 

a.     3.43213 
—  4  14  15  .   Ar.  CO.  cos.  0.00119 


X         ...     45  12  -  2712"  -  log.  3.43332 
h        '        •    -  37   0  34  -    -   -  tan.  9.87725 

c.  3.31057 


PROB.  XVII.    TO  FIND  THE  MOOn's  APPAR.  LONG.  AND  LAT.    295 

c.     3.31057 
K    -   -   -  47°  24'  44"  -   -   -  sin.  9.86701 


M    -    -   -     25   5  -  1505"  -  lo?.  3.17758 


c.  3.31057 
K  +  ?f-    -    -  47  49  49  -   -   -  sin.  9.86991 


w'    -    -    -     25  15  -  1515".2  -  log.  3.18048 


K+  w' 

-    47 

49 

59    - 

- 

sin.  9.86993 

P 

- 

25 

15.3 

1515".3  - 

log.  3.18050 

True  long.  - 

-  247 

37 

6.7 

Appar.  long. 

-  248 

2 

22.0 

P 

- 

• 

- 

log.  3.18050 

■K  —  a;- 

-_4 

59 

27    - 

- 

sin.  8.93957— 

X 

- 

- 

- 

Ar.  CO. 

,  cos.  0.00119 

u         -         - 

- 

- 

- 

Ar.  CO. 

,  log.  6.82242 

X' 

-—5 

1 

10  - 

- 

tan.  8.94368— 

X' 

-      5 

1 

10    - 

- 

cos.  9.99833 

44 

54.4 

2694".4  - 

a.    3.43213 

V 

lofif.  3.43046 

h 

- 

- 

- 

- 

tan.  9.87725 

X' 

- 

- 

- 

- 

tan.  8.94368— 

K  +  ip      - 

-    47 

37 

22    - 

- 

cos.  9.82867 

z         -        - 

- 

—  2 

0.2 

120".2  - 

log.  2.08006- 

V  —  z 

- 

46 

54.6 

V  —  z  (sign  changed)    —  46     54.6 
True  lat.       -  —  4     14     14.7 


Appar.  lat.    -        -       5       I       9.3  S. 

p log.  3.18050 

X  .        -        .        .        .         -        Ar.  CO.  cos.  0.00119 

u Ar.  CO.  log.  6.82242 

X' cos.  9.99833 

R        -        -        -  15     24.2      924".2  -  log.  2.96577 


Augra.  semi-diam.  15    29.4      929".4  -  log.  2.96821 


296 


ASTRONOMT. 


Exam.  2.  About  the  middle  of  the  eclipse  of  the  sun  on  the 
18th  of  September,  1838,  the  moon's  longitude  was  175°  29' 
19".0,  latitude  47'  47".5,  equatorial  parallax  53'  53  ".5,  and  semi- 
diameter  14'  41". 1 ;  and  the  longitude  of  the  nonagesimal  at 
New  York  was  216°  20'  50",  the  altitude  32°  15'  48" :  required 
the  apparent  longitude  and  latitude,  and  the  augmented  semi- 
diameter  of  the  moon. 


Equat.  paral.    53'  53".5 
Reduction,  4 .4 


Moon's  long.      175°  29'  19" 
Long,  nonag.     216    20    50 


Sun's  paral. 

53  49.1           K  =  - 

8.6           h  = 

-40  51  31 

32  15  48 

P  = 
P    .    - 
h 

X 

=  53  40.5           X  = 
-  3220".5  - 

32°  15' 48"   - 

-  47  47.5  -    -  Ar.  co 

-  45  23.5  -  2723  ".5  - 
32  15  48 

—40  51  31 

—  18  45    -  1125"  - 

—  41  10  16 

—  18  52.9  -  1132".9- 

—  41  10  24 

—  18  52.9  -  1132  ".9- 
175  29  19.0 

0  47  47.5 
log.  3.50792 
COS.  9.92716 
1.  COS.  0.00004 

X 

h 

log.  3.43512 
tan.  9.80023 

K   - 

c.   3.23535 
sin.  9.81570 — 

u 

log.  3.05105— 

K  +  M 

c.   3.23535 
sin.  9.81844— 

v! 

log.^3.05379— 

K  +  m' 

c.  3.23535 
sin.  9.81844— 

True  long. 

log.  3.05379— 

Appar.  long. 

175  10  26.1 

PROB.  XVIII.  TO  FIND  A  STAR's  MEAN  R.  ASC.  AND  DEC.       297 


X 
u 
X — ■  X 


2'  24".0 


-  log.  3.05379 

Ar.  CO.  COS.  0.00004 

Ar.  CO.  log.  6.94895 

144".0  -        -  log.  2.15836 


Appar.  latitude     2'  24".9  N.      144".9  -        -    log.  2.16114 


P 

X 

u 
R 


log.  3.05379 

Ar.  CO.  cos.  0.00004 

Ar.  CO.  log.  6.94895 

14'  41".l  -     881".l  -        -  loo-.  2.9450^ 


Augm.semi-diam.  14  46.7    -    886".7   -        -  log.  2.94780 


PROBLEM  XVIII. 

To  find  the  Mean  Right  Asceyision  and  Declination .^  or  Longi- 
tude and  Latitude  of  a  Star,  for  a  given  titne,  from  the 
Tables. 

Take  the  difference  between  the  given  year  and  1840.  Then 
seek  in  Table  XV  for  the  fraction  of  the  year  answering  to  the 
given  month  and  days,  and  add  it  to  this  difference,  if  the  given 
time  is  after  the  beginning  of  the  year  1840  ;  but  if  it  is  before, 
subtract  it.  Multiply  the  sum  or  difference  by  the  annual 
variation  given  in  the  catalogue  (Table  XC),  and  the  product 
will  be  the  variation  in  the  interval  between  the  given  time  and 
the  epoch  of  the  catalogue.  Apply  this  product  to  the  quantity 
given  in  the  catalogue,  according  to  its  sign,  if  the  given  time  is 
after  the  beginning  of  the  year  1840,  but  with  the  opposite  sign 
if  it  is  before,  and  the  result  will  be  the  quantity  sousfht. 

Exam.  1.  Required  the  mean  right  ascension  and  declination 
of  the  star  Sirius  on  the  15th  of  August,  1842. 
Interval  between  given  time  and  beginn.  of  1840  (t),     2.619  yrs. 
Annual  variation  of  right  ascension,        -         -         -         2.646s, 


Variation  of  right  ascension  for  interval  t,        -         -       6.93s. 

A  similar  operation  gives  for  the  variation  of  declination  in 
the  same  interval,  11". 65. 
38 


298  ASTRONOMY. 

Mean  right  ascen.  beginning  of  1840,  Table  XC,    6^-  38™- 5.76= 
Variation  for  interval  t. -H  6.93 


Mean  right  ascension  required,  -         -         -     6  38  12.69 

Mean  declination  beginning  of  1840,          -  16"  30'   4".79  S. 

Variation  for  interval  t, +  11  .65 

Mean  declination  reqnired,        -         -         -  16  30  16.44S. 

2.  Required  the  mean  longitude  and  latitude  of  Aldebaran  on 
the  20th  of  October,  1838. 

Interval  between  given  time  and  begin,  of  1840,  {t)      1.200  yrs. 
Annual  variation  of  longitude,  .         _         .  50".208 

Variation  of  longitude  for  interval  t,  -         -  60".2 

A  similar  operation  gives  for  the  variation  of  latitude  in  the 
same  interval  0".4. 

Mean  longitude  beginning  of  1840,           -  2^-  7°  33'  5".4 

Variation  for  interval  t, —  1    0  .2 


Mean  longitude  required,  -        -        -        2    7    32  5  .2 

Mean  latitude  beginning  of  1840,      -        -        -     5°  28'  38''.0  S. 
Variation  for  interval  t,  -         -         -        -         -  +0-4 


Mean  latitude  required, 5    28  38.4  S. 

3.  Required  the  mean  right  ascension  and  declination  of  Ca- 
pella  on  the  9th  of  February,  1839  '? 

Ans.  Mean  right  ascension  5''-  4"»-  48.74s  ^  and  mean  declina- 
tion 45°  49'  38".53  N. 

4.  Required  the  mean  longitude  and  latitude  of  Aldebaran  on 
the  16th  of  April,  1845  ? 

Ans.  Mean  longitude  2"-  7°  37'  30".9,  and  mean  latitude  5°  28' 
36".2. 


PROB.  XIX.    TO  FIND  A  STAR's  ABERR.   IN  R.  ASCEN.,  &C.      299 


PROBLEM    XIX. 

To  find  the  Aberration  of  a  Star  in  Right  Ascension  atid  Decli- 
tiation,  for  a  given  Day. 

This  problem  may  be  resolved  for  any  of  the  stars  in  the 
catalogue  of  Table  XC  by  means  of  the  following  formulae  : 

log.  (aber.  in  right  ascen.)  =  log.  M  +  log.  sin.  (O  +  <p)  —  10. 

log.  (aber.  in  declin.)  =■  log.  N  +  log.  sin.  ( O  +  ^)  —  10, 
in  which  M,  N,  are  constants,  O  the  longitude  of  the  sun  on  the 
given  day,  and  9,  ^,  auxiliary  angles.  Log  M,  log  N,  and  the 
angles  9,  ^,  are  given  for  each  of  the  stars  in  the  catalogue,  in 
Table  XCL  O  may  be  derived  from  an  ephemeris  of  the  sun, 
or  it  may  be  computed  from  the  solar  tables  by  Problem  IX. 

Exam.  1.  What  was  the  amount  of  aberration,  in  right  ascen- 
sion and  declination  of  a  Orionis  on  the  20th  of  December, 
1837,  the  sun's  longitude  on  that  day  being  8^-  28°  28'  ? 

Right  Ascension. 
Table  XCI,  9  -         6^-  3°  13'      log.  M        -        0.1361 

O        -         8  28   28 


0  +  9  3     1   41        -        -        sin.  9.9998 


Aberration  =  1".37     -        -        -        -        log.  0.1359 

Declination. 
Table  XCI,    ^         -        8^-28"  23'       log.  N        -        0.7521 

G        -        8   28    28 


0  +  ^  -        5   26    51         -        -        sin.  8.7399 


Aberration  =  0".31    -        -        -        -         log.  1.4920 
2.  Required  the  aberrations  in  right  ascension  and  declina- 
tion of  a  Andromedse  on  the  1st  of  May,  1838,  the  sun's  longi- 
tude being  1«- 10°  38'. 

Ans.  Aberr.  in  right  ascension  — 1".07,  and  aberr.  in  declina- 
tion —  11' .70. 


300  ASTRONOMY. 


PROBLEM     XX. 

To  find  the  Nutation  of  a  Star  in  Right  Ascension  and  Decli- 
nation, for  a  given  Day. 

This  Problem  may  be  solved  by  means  of  the  formulee, 

locr.  (nuta.  in  right  asc.)  =  log.  M'  +  log.  sin  (Q  +9')  —  10 ; 

log.  (nuta.  in  declinat.)  =  log.  N'  -j-  log.  sin  (Q  +  ^')  —  10  ; 
in  which  M',  N'  are  constants,  Q  the  mean  longitude  of  the 
moon's  ascending  node,  and  ©',  &'  auxiliary  angles.  Log.  M',  log. 
N',  and  the  angles  9',  d'  are  given  for  each  of  the  stars  in  the  cata- 
logue, in  Table  XCL  The  mean  longitude  of  the  moon's  ascend- 
ing node  is  given  for  every  tenth  day  of  the  year  in  the  Nautical 
Almanac,  page  266,  and  may  be  easily  found  for  any  interme- 
diate day  from  the  daily  motion  inserted  at  the  foot  of  the  column 
of  longitudes.  It  may  also  be  had  by  finding  the  supplement  of 
the  moon's  node,  for  the  given  time,  from  the  tables,  and  subtract- 
ing it  from  12=--  0°  7'. 

Exam.  1.  What  was  the  amount  of  the  nutation,  in  right  ascen- 
sion and  declination,  of  a  Orionis  on  the  20th  of  December,  1837, 
the  mean  longitude  of  the  moon's  node  on  that  day  being  18°  54'  1 
Right  Ascension. 

TableXCI,  9'        -    6«-     0°  15',     log.  M'   -        -    0.0481    \ 
Q        -    0     18     54 


S  +  <?'     6     19       9  - 

sin. 
log.' 

sin. 
log. 

9.5159- 

Nutation  =  —0".  37     - 

1.5640- 

Declination. 
Table  XCIjd'         -    3«-     2°  37',     log.  N'  - 
Q        -     0     18     54 

0.9657 

8  +d'      3     21     31  - 

9.9686 

Nutation  =       8".60     - 

0.9343 

2.  Required  the  nutations  in  right  ascension  and  declination  of 
a  Andromedae  on  the  1st  of  May,  1838. 

Ans.  Nutation  in  right  ascension  —  0".54,  and  nutation  in 
declination  —  1".43. 


PROB.  XX.  TO  FIND  A  STAR's  NUTAT.  IN  R.  ASC,  &.C.        301 

Note,  When  the  apparent  place  of  a  star  is  desired  with  great 
accuracy,  the  ijolar  nutations  must  also  be  estimated  and  allowed 
for.  These  may  be  determined  by  repeating  the  process  for  find- 
ing the  lunar  nutations,  only  using  twice  the  sun's  longitude  in 
place  of  the  longitude  of  the  moon's  node,  and  multiplying  the  re- 
sults by  the  decimal  .075. 

The  calculation  of  the  solar  nutations  in  Example  1st,  is  as 
follows  : 

Right  Ascension. 


Table  XCI,    9'       -     6«-     0°   15',     log.M'     - 
20     -     4     56     56 

-      0.0481 

20+9'  10    57     11    - 

—  0".06 
.075 

sin.  9.7455— 
logri.7936— 

Solar  Nutat.  =— O'.OO 
Declination. 
Table  XCI,     6'     -    3«-     2°  37',     log.  N'    - 
2  0   -     4     56     56 

-      0.9657 

7     59     33   - 

sin.  9.9999— 

—  9".24       - 
.075 

-      0.9656— 

Solar  Nutat.  =  —  0  .69 
In  Example  2d,  we  find  for  the  solar  nutation  in  right  ascen- 
sion, —  0".08,  and  for  the  solar  nutation  in  declination,  —  0".57. 


"-      PROBLEM    XXI.  '      ' 

To  find  the  Apparent  Right  Asceyision  and  Declination  of  a 
Star  on  a  given  Day. 
Find  the  mean  right  ascension  and  declination  for  the  given 
day  by  Problem  XVIII ;  then  compute  the  aberrations  in  right 
ascension  and  declination  by  Problem  XIX,  and  the  lunar  and 
solar  nutations  in  right  ascension  and  declination  by  Problem 


302  ASTRONOMY. 

XX.  Apply  the  aberrations  and  nutations  according  to  their 
signs,  to  the  mean  right  ascension  and  declination  on  the  given 
day,  observing  that  the  declination  when  south  is  to  be  marked 
negative,  and  the  results  will  be  the  apparent  right  ascension  mid 
declination  sought. 

Exam.  1.  What  was  the  apparent  right  ascension  and  decli- 
nation of  a  Orionis  on  the  20th  of  Docember,  1837  ? 

h.   m.         s.  °        '        " 

Table  XC,  M.  right  ascen.    5  46  30.71,    M.  dec.  7  22  17.14N. 
Variations        -  —  6.59       -        -         —  2.42 


5  46  24.12 

7  22  14.72 

Aberr.     - 

+ 1-37      - 

+  0.31 

Lun.  nutat. 

—  0.37      - 

+  8.60 

Sol.  nutat. 

0.00      - 

—  0.69 

App.  right  asc.     5  46  25.12,  App.dec.  7  22  22.94N. 

2.  Required  the  apparent  right  ascension  and  declination  of  a 
Andromedae  on  the  1st  of  May,  1838. 

Ans.  Appar.  right  ascen.  Oh.  Om.  0.90s,,  and  appar.  dec.  28° 
11'  39"  90. 


PROBLEM    XXII. 

To  find  the  Aberration  of  a  Star  in  Longitude  and  Latitude, 

for  a  given  Day. 
The  formulae  for  the  computation  are, 

log.  (aber.  in  long  )  =  1.30880  +  log.  cos.  (6s.  +  O  —  L)  +  ar. 
CO.  log.  COS.  X  —  10  ; 

log.  (aber.  in   lat.)  =  1.30880  +  log.  sin.  (6s.  +  O  —  L)  +log. 

sin.  X  —  20 ; 

in  which  O  =  longitude  of  the  sun  on  the  given  day  ;  L  =  mean 
longitude  of  the  star  ;  and  X  =  mean  latitude  of  the  star. 

Exam.  1.  Required  the  aberrations  in  longitude  and  latitude 
of  a.  ScorpiL,  on  the  26th  of  February,  1838.  the  sun^s  longitude 
on  that  day  being  11^-  7°  29'. 


PROB.  XXII.    TO  FIND  A  STAR's  ABERR.   IN  LONG.,  &C.        303 

By  Prob.  XVIII,  L  =  8^-  7°  30',       and  X  =  4°  32'  S. 

6s.  +  o      -    17     7    29         Const,  lo^.  1.3088 


6s.  +  O  —  L  8  29    59 

-    COS.  6.4637— 

X        -        -           4   32 

Ar.  co.cos.  0.0014 

Aberr.  in  long.  =  —  0".00  log.  3.7739— 

Const,  log.  1.3088 
6s.  +  o  —  L    8^-  29°  59'     -  sin.  9.2474— 

X    -        -        -         4   32     -  sin.  9.7099 


Aberr.  in  lat.  =  —  1".61,  log.  0.2661— 
2.  Required  the  aberrations  in  longitude  and  latitude  of  Arc- 
turus  on  the  5th  of  October,  1838,  the  sun's  longitude  being 
6^-  11°  47'. 

Ans.  Aberr.  in  long.  —  23".34,  and  aberr.  in  lat.  1".85. 
Note.  The  nutation  in  longitude  of  a  fixed  star  may  be  found 
after  the  same  manner  as  the  nutation  in  longitude  of  the  sun. 
(See  Problem  IX,  page  271). 


PROBLEM   XXIII. 

To  find  the  Apparent  Longitude  and  Latitude  of  a  Star,  for  a 

given  Day. 

Find  the  mean  longitude  and  latitude  on  the  given  day  by 
Problem  XVIII.  Find  also  the  aberrations  in  longitude  and  lati- 
tude by  Problem  XXII,  and  the  nutation  in  longitude,  as  in 
Problem  IX.  Apply  the  aberration  and  nutation  in  longitude, 
according  to  their  signs,  to  the  mean  longitude,  and  the  result  will 
be  the  apparent  longitude  ;  and  apply  the  aberration  in  latitude 
according  to  its  sign,  to  the  mean  latitude,  and  the  result  will  be 
the  apparent  latitude. 

Exam.  1.  Required  the  apparent  longitude  and  latitude  of  a 
Scorpii  on  the  26th  of  February,  1838. 


304  ASTRONOMY. 

Table  XC,  M.  long.    8-  7°  31'  45".2,      M.  lat.  4°  32'    51".6  S. 
Var.  —  1    32  .57  0  .78 


8    7     30    12  .63 

Aberr. 

0  .00 

Nutat. 

—  4  .40 

4    32    50.82 
—  1  .61 


App.long.8     7    30       8  .23,  App.lat.  4    32    49  .21S. 

2.  Required  the  apparent  longitude  and  latitude  of  Arcturus  on 
the  5th  of  October,  1838. 

Ans.  Appar.  long.  6^-  21<=  58'  40".4,  and  appar.  lat.  30^  51 
19".l. 


PROBLEM    XXIV. 

To  compute  the  Longitude  mid  Latitude  of  a  Heavenly  Body 

from  its  Right  Ascension  and  Declination,  the  Obliquity  of 

the  Ecliptic  being  given. 

This  Problem  may  be  solved  by  means   of  the  following 
formulae : 

log.  tang:?;=log.  tang  D  +  ar.  co.  log.  sin  R; 

log.  tangL  =  log.  cos.  (.r — w)  +  log.  tarigR  -f  ar.  co.  log.  cos.  a: — 10 ; 

log.  tang  X— log.  tang  [x  —  u)  -\-  log.  sin  L  —  10 ; 
in  which 

R  =  the  Right  Ascension  ; 

D  =  the  Declination  (minus  when  south) ; 

L  =  the  Longitude ; 

X  =  the  Latitude  ; 

cd  =  the  Obliquity  of  the  ecliptic  ; 
X  is  an  auxiliary  arc.  It  must  be  taken  according  to  the  sign  of 
its  tangent,  but  always  less  than  180°.  The  longitude  will 
always  be  in  the  same  quadrant  as  the  right  ascension.  The 
latitude  must  be  taken  less  than  90°,  and  will  be  north  or  south, 
according  as  the  sign  is  positive  or  negative. 

Note.  When  the  mean  longitude  and  latitude  are  to  be  derived 
from  the  mean  right  ascension  and  declination,  the  mean  obli- 


PROB.  XXV.  TO  COMPUTE  THE  R.  ASC.  AND  DEC.  OF  A  BODY.      305 

quity  of  the  ecliptic  is  taken.  When  the  apparent  longitude  and 
latitude  are  to  be  derived  from  the  apparent  right  ascension  and 
declination,  found  as  in  Problem  XXI,  the  apparent  obliquiiy  is 
taken.  The  mean  obliquity  of  the  ecliptic  at  any  assumed  time 
is  easily  deduced  from  Table  XXII.  The  apparent  obliquity  is 
found  by  Problem  X. 

Exam.  1.  On  the  20th  of  June,  1838,  the  right  ascension  of 
Capella  was  76^  11'  29",  the  declination  45°  49'  35"  N.,  and  the 
obliquity  of  the  ecliptic  23°  27'  37" :  required  the  longitude  and 
latitude. 

D  =  45°  49'  35"     -        -        -  tan.  0.0125295 

R=  76    1129      -        -        Ar.  CO.  sin.  0.0127367 


a;  =  46 
w  =23 

39  54   - 

27  37 

tan.  0.0252662 

a:  — co  =  23 
R  =  76 
X  =46 

12  17      - 

11  29   - 
39  54   - 

36  3   - 

36  3   - 

12  17   - 

cos.  9.9633641 

tan.  0.6094483 

Ar.  CO.  COS.  0.1635095 

Long.  =  79 

tan.  0.73G3219 

L  =79 
X  —  w  =23 

sin.  9.9928071 
tan.  9.6321516 

Lat.  =  22    51  47      -        -        -  tan.  9.6249587 

2.  Given  the  right  ascension  of  ISpica  199°  11'  35",  and  decli- 
nation 10°  19'  24"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  36", 
on  the  1st  of  January,  1840,  to  find  the  longitude  and  latitude. 
Ans.  Long.  201°  36'  32",  and  lat.  2°  2'  30"  S. 


PROBLEM    XXV. 

To  co?npnte  the  Right  Ascension  and  Declination  of  a  Hea- 
venly  Body  from  its  longitude  and  latitude,  the  obliquity 
of  the  ecliptic  being  given. 
The  formulae  for  the  solution  of  this  problem  are, 

log.  tang  y  =  log.  tang  X  -f  ar.  co.  log.  sin  L ; 
39 


306  ASTRONOMT. 

log.  tang  R  =  log.  COS.  (y  4"'^)  + log-  tangL-j-ai*-  co.  log.  cos.  y  — 10; 
log.  tang  D  =  log.  tang  (y  +  w )  +  log.  sin.  R  — 10 ; 
in  which 

L  =  the  Longitude ; 

X  =  the  Latitude  [minus  when  South) ; 

R  =  the  Right  Ascension  ; 

D  =  the  Declination  ; 

w  =  the  Obliquity  of  the  ecliptic  ; 
y  is  an  auxiliary  arc.  It  must  be  taken  according  to  the  sign  of 
its  tangent,  but  always  less  than  180°.  The  right  ascension  will 
always  be  in  the  same  quadrant  with  the  longitude.  The  decli- 
nation must  be  taken  less  than  90°,  and  will  be  north  or  south, 
according  as  the  sign  is  positive  or  negative. 

Note.  The  mean  or  apparent  obliquity  of  the  ecliptic  is  taken, 
according  as  the  given  and  required  elements  are  mean  or 
apparent. 

Exam  1.  On  the  1st  of  January,  1830,  the  longitude  o{  Sirius 
was  3^-  11°  44'  18",  the  latitude  39°  34'  1"  S.,  and  the  obliquity 
of  the  ecliptic  23°  27'  41" :  required  the  right  ascension  and 
declination. 

X  =  _  39°  34'    1"       -        -  tan.  9.9171381  — 

L  =     101    44  18        -       Ar.  co.  sin.  0.0091788 


2/=     139    50  14        -        -          tan.  9.9263169 — 
w  =       23    27  41  


y  +  £j  =  163  17  55  -  -          cos.  9.9812819 — 

L  =  101  44  18  -  -          tan.  0.6823798  — 

y=  139  50  14  -  Ar.  CO.  COS.  0.1167843  — 

Riffht  ascen.  =  99  24  48  -  -          tan.  0.7804460  — 


R=       99    24  48        -        -  sin.  9.9941121 

y  +  w  =     163    17  55        -        -  tan.  9.4771803— 

Dec.  =       16    29  20  S.  -  tan.  9.4712924  — 

2.  Given  the  longitude  of  Aldebaran  67°  33'  5",  and  latitude 

5°  28'  38"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  36",  on  the 

1st  of  January,  1840.  to  find  the  right  ascension  and  declination. 

Ans.  Right  ascension  66°  41'  4",  and  declination  16°  10'  57"  N. 


PROB.  XXVI.    TO  FIND  THE  ANGLE  OF  POSITION  OF  A  BODY.  307 


PROBLEM    XXVI. 

The  Longitude  and  Declination  of  a  Body  being  giveii,  and 
also  the  Ohliquity  of  the  Ecliptic^    to  find   the   Angle   of 
Position. 
The  formula  is 

log.  sin  p  =  log.  sin  w  +  log.  cos.  L  +  ar.  co.  log.  cos.  D  — 10 ; 

p  =  Angle  of  Position  (required) ; 

L  =  Longitude ; 

D  =  Declination ; 

If  =  Obliquity  of  the  ecliptic. 

The  angle  of  position  p  must  be  taken  less  than  90°.  It  is  to 
be  observed  also  that  when  the  longitude  is  less  than  90°,  or 
more  than  270°,  the  northern  part  of  the  circle  of  latitude  lies  to 
the  west  of  the  circle  of  declination,  but  that  when  the  longitude 
is  between  90°  and  270°,  it  lies  to  the  east. 

Note.  The  angle  of  position  may  also  be  computed  from  the 
right  ascension  and  latitude,  by  means  of  a  formula  similar  to 
that  just  given,  namely, 

log.  sin  p  =  log.  sin  w  +  log.  cos.  R  +  ar.  co.  log.  cos.  X — 10. 

Exam.  1.  Given  the  longitude  of  Regulus  147°  27'  54",  and 
declination  12°  47'  45"  N.,  and  the  obliquity  of  the  ecliptic  23° 
27'  41",  to  find  the  angle  of  position. 

w  =  23°  27'  41"  -  -  sin.  9.6000260 
L  =  147  27  54  -  -  cos.  9.9258601 
D=    12   47  45       -      Ar.  CO.  cos.  0.0109217 


Angle  of  pos.  =    20     7  58       -        -        sin.  9.5368078 

The  circle  of  latitude  lies  to  the  east  of  the  circle  of  decli- 
nation. 

2.  Given  the  longitude  of  Fomalhaut  351°  51'  38",  and  decli- 
nation 30°  31'  14"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  41", 
to  find  the  angle  of  position.  Ans.  27°  13'  36". 

The  circle  of  latitude  lies  to  the  west  of  the  circle  of  decli- 
nation. 


308 


ASTRONOMY. 


PROBLEM    XXVII. 

To  find  from  the  Tables  the  Time  of  Neiv  or  Full  Moon,  for 
a  given  Year  and  Month. 

For  the  New  Moon. 

Take  from  TableLXXXVI,  the  time  of  mean  new  moon  in  Jan- 
uary, and  the  Arguments  I,  II,  III  and  IV,  for  the  given  year. 
Take  from  Table  LXXXVII,  as  many  Umations  with  the  corres- 
ponding variations  of  Arguments  I,  II,  III,  and  IV,  as  the  given 
month  is  months  past  January,  and  add  these  quantities  to  the 
former,  rejecting  the  ten  thousands  from  the  sums  in  the  columns 
of  the  first  two  arguments,  and  the  hundreds  from  the  sums  in 
the  columns  of  the  other  two.  Seek  the  number  of  days  from  the 
first  of  January  to  the  first  of  the  given  month,  in  the  second  or 
third  column  of  Table  LXXXVIII,  according  as  the  given  year  is 
a  common  or  bissextile  year,  and  subtract  it  from  the  sum  in  the 
column  of  mean  new  moon  :  the  remainder  will  be  tabular  time 
of  mean  new  moon  for  the  given  month.  It  will  sometimes 
happen  that  the  number  of  days  taken  from  Table  LXXXVIII, 
will  exceed  the  number  of  days  of  the  sum  in  the  column  of 
mean  new  moan  :  in  this  case  one  lunation  more,  with  the  cor- 
responding arguments,  must  be  added. 

With  the  sums  in  the  columns  I,  II,  III,  and  IV,  as  arguments, 
take  the  corresponding  equations  from  Table  LXXXIX,  and  add 
them  to  the  time  of  mean  new  moon  :  the  sum  will  be  Approxi- 
mate time  of  new  moon  for  the  given  month,  expressed  in  mean 
time  at  Greenwich. 

Next,  for  the  approximate  time  of  new  moon  calculate  the  true 
longitudes  and  hourly  motions  in  longitude  of  the  sun  and  moon  ; 
subtract  the  less  longitude  from  the  greater,  and  the  hourly  mo- 
tion of  the  sun  from  the  hourly  motion  of  the  moon  ;  and  say,  as 
the  difference  between  the  hourly  motions  :  the  difference  between 
the  longitudes  :  :  60  minutes  :  the  correction  of  the  approximate 
time.  The  correction  added  to  the  approximate  time,  when  the 
sun's  longitude  is  greater  than  the  moon's,  but  subtracted ,  when 
it  is  less^  will  give  the  true  time  of  new  moon  required,  in  mean 


PROB.  XXVII.  TO  FIND  THE  TIME  OF  NEW  OR  FULL  MOON.    309 

time  at  Greenwich.     This  time  may  be  reduced  to  the  meridian 
of  any  given  place  by  Problem  V. 

For  Full  Moon. 

Take  from  Table  LXXXVI,  the  time  of  mean  new  moon,  and 
the  corresponding  Arguments  I,  II,  III,  and  IV,  for  January  of  the 
given  year,  and  from  Table  LXXXVII,  a  half  lunation  with  the 
corresponding  changes  of  the  arguments.  Then,  when  the  time 
of  mean  new  moon  for  January  is  on  or  after  the  16th,  subtract 
the  latter  quantities  from  the  former,  increasing,  when  necessary 
to  render  the  subtraction  possible,  either  or  both  of  the  first  two 
arguments  by  10,000,  and  of  the  last  two  by  100  ;  but  add  them 
when  the  time  is  before  the  16th.  The  result  will  be  the  tabular 
time  of  mean  full  moon  and  the  corresponding  arguments,  for 
January.  Proceed  to  find  the  approximate  time  of  full  moon 
after  the  same  manner  as  directed  for  the  new  moon. 

For  the  approximate  of  full  moon  calculate  the  true  longi- 
tudes and  hourly  motions  in  longitude  of  the  sun  and  moon. 
Subtract  the  sun's  lonoritude  from  the  moon's,  adding  360°  to  the 
latter  if  necessary.  Take  the  difference  between  the  remainder 
and  VI  signs,  and  call  the  result  R.  Also  subtract  the  hourly 
motion  of  the  sun  from  the  hourly  motion  of  the  moon.  Then 
say,  as  the  difference  between  the  hourly  motions  :  R  :  :  60m. 
:  the  correction  of  the  approximate  time.  The  correction  added 
to  the  approximate  time  of  full  moon,  when  the  excess  of  the 
moon's  longitude  over  the  sun's  is  less  than  VI  signs,  but  subtract- 
ed when  it  is  greater,  will  give  the  true  time  of  full  moon. 

Exam.  1.  Required  the  time  of  new  moon  in  September, 
1838,  expressed  in  mean  time  at  New- York. 


1838, 
8  lun. 

M.  New  Moon. 

I. 

II. 

III. 

IV. 

d.  h.  m. 

24  16  .53 

236  5  f-2 

0G31 
6468 

917.5 
5737 

4912 

99 
22 

21 

85 
93 

78 

Days, 

260  22  45 
243 

7149 

Sept'r, 

I. 

11. 

in. 

IV. 

17  22  45 

0  16 

9  35 

3 

10 

Sept'r, 

18  8  49 

Appro: 

cimato  t 

tme. 

310 


ASTRONOMY. 


Moon's  true  long,  found  for  approx.  time,  is   5«-  25°  29'    19" 
Sun's  do.  do.  do.  5    25    27    27 

Difference,        ------ 


Moon's  hourly  motion  in  long,  is 
Sun's  do.  do. 


Difference, 

As  27'  1" :  1'  52' 


60™  :  4c^  9^-,  the  correction. 


1  52 

29  28 

2  27 

27  1 


Approx.  time  of  new  moon,  September, 
Correction,       ----- 

True  time,  in  mean  time  at  Greenwich, 
Diff.  of  meridians,     -         -         -         - 


Igd     8h- 


49m.  09 
-4     9 


18     8 


44  51 
56     4 


True  time,  in  mean  time  at  New  York,      -     18     3 

Exam.  2.  Required  the  time  of  full  moon  in  April, 
pressed  in  mean  time  at  New  York. 


48  47 
1838,  ex- 


1838, 
ilun. 

M.  Full  Moon. 

I. 

II.    j  III. 

IV. 

d.    h.  m. 
24  16  53 
14  18  22 

0681 
404 

9175      99 
5359      58 

85 
50 

3Iun. 

9  22  31 

88  14  12 

0277 
2425 

3316 
2151 

41 

46 

35 
97 

Days, 

98  12  43 

93 

2702 

5967 

87 

32 

April, 

I. 

II. 

HI. 

IV. 

8  12  43 

8  29 

16     7 

15 

30 

April, 

9  14    4 

Approj 

cimate  t 

ime. 

Moon's  true  long,  found  for  approx.  time,  is  6^-   19° 
Sun's  do.  do.  do.  0     19 


44'    17" 
45     22 


29 
0 


58 
0 


55 

0 


R 


PROB.  XXVIII.  TO  FIND  THE  NUMB.  OF  ECLIPSES  IN  A  YEAR.    311 

Moon's  hourly  motion  in  long,  is  -         -     30'  15" 
Sun's  do.  do.  .        .      2  27 


Difference -     27  48 

As  27' 48"  :  1'  5" : :  GO'"-  :  2'"-  20^-,  the  correction. 

Approximate  time  of  full  moon,  April,  9"^- 14^*-  4'"-  0'- 

Correction, +  2  20 

True  time,  in  mean  time  at  Greenwich,         9    14     6  20 
Ditf.  of  meridians,      .         -         -         -  4  56     4 

True  time,  in  mean  time  at  New  York,         9      9  10  16 

3.  Required  the  time  of  new  moon  in  September,  1837,  ex- 
pressed in  mean  time  at  Philadelphia ;  taking  the  longitudes  for 
the  approximate  time  from  the  Nautical  Almanac. 

Ans.  29d.  3h.  Om.  5s. 

4.  Required  the  time  of  full  moon,  in  October,  1837,  ex- 
pressed in  mean  time  at  Boston. 

Ans.  13d.  6h.  30m.  25s. 


PROBLEM    XXVIII. 

To  determine  the  mimher  of  Eclipses  of  the  Sun  and  Moon 
that  may  he  expected  to  occur  in  any  given  Year,  and  the 
Times  nearly  at  which  they  ivill  take  place. 

For  the  Eclipses  of  the  Sun. 

Take,  for  the  given  year,  from  Table  LXXXVI  the  time  of 
mean  new  moon  in  January,  the  arguments  and  the  number  N. 
If  the  number  N  differs  less  than  37  from  either  0,  500.  or  lOOO, 
an  eclipse  must  occur  at  that  new  moon.  If  the  difference  is 
between  37  and  53,  there  may  be  an  eclipse,  but  it  is  doubtful, 
and  the  doubt  can  only^  be  removed  by  a  calculation  of  the  true 
places  of  the  moon  and  sun.  If  the  difference  exceeds  53,  an 
eclipse  is  impossible. 

If  an  eclipse  may  or  must  occur  at  the  new  moon  in  January, 
calculate  the  approximate  time  of  new  moon  by  Problem  XXVII, 
and  it  will  be  the  time  nearly  of  the  middle  of  the  eclipse,  ex- 


312  ASTRONUMV. 

pressed  in  mean  time  at  Greenwich.  This  may  be  reduced  to 
the  meridian  of  any  other  place  by  Problem  V. 

To  find  the  first  new  moon  after  January,  at  which  an  eclipse 
of  the  sun  may  be  expected,  seek  in  column  N  of  Table  LXXXVII 
the  first  number  after  that  answering  to  the  half  lunation,  that, 
added  to  the  number  N  for  the  given  year,  will  make  the  sum 
come  within  53  of  0,  500,  or  1000.  Take  the  corresponding  lu- 
nations, changes  of  the  arguments,  and  the  number  N,  and  add 
them,  respectively,  to  the  mean  new  moon  in  January,  the  argu- 
ments, and  the  number  N,  for  the  given  year.  Take  from  the 
second  or  third  column  of  Table  LXXXVIII,  according  as  the 
given  year  is  a  common  or  bissextile  year,  the  number  of  days 
next  less  than  the  days  of  the  sum  in  the  column  of  mean  new 
moon,  and  subtract  it  from  this  sum;  the  remainder  will  be  the 
tabular  time  of  mean  new  moon  in  the  month  corresponding  to 
the  days  taken  from  Table  LXXXVIII.  At  this  new  moon  there 
maybe  an  eclipse  of  the  sun ;  and  if  the  sum  in  the  column  X  is 
within  37  of  the  numbers  mentioned  above,  there  must  be  one. 
Find  tlie  approximate  time  of  new  moon,  and  it  will  be  the  time 
nearly  of  the  middle  of  the  eclipse. 

If  any  of  the  other  numbers  in  the  last  column  of  Table 
LXXXVII  are  found,  when  added  to  the  number  N  of  the  given 
year,  to  give  a  sum  that  falls  within  the  limit  53,  proceed  in  a 
similar  manner  to  find  the  approximate  times  of  the  eclipses. 

Note.  "When  the  sum  of  the  numbers  N,  or  the  number  N 
itself,  in  case  the  eclipse  happens  in  January,  is  a  little  above  0, 
or  a  little  less  than  500,  the  moon  will  be  to  the  north  of  the  sun, 
and  there  is  a  probabilitt/  that  the  eclipse  will  be  visible  at  any 
given  place  in  north  latitude  at  which  the  approximate  time  of 
the  eclipse  found  as  just  explained  and  reduced  to  the  meridian 
of  the  place  comes  during  the  day-time.  When  the  number  N 
found  for  the  eclipse  is  more  than  500,  the  moon  will  be  to  the 
south  of  the  sun,  and  the  eclipse  will  seldom  be  visible  in  the 
northern  hemisphere  except  near  the  equator. 

For  the  Eclipses  of  the  Moon. 
Find  the  time  of  full  moon  and  the  corresponding  arguments 
and  number  N,  for  January  of  the  given  year,  as  explained  in 
Problem  XXVII.     Then   proceed  to  find  the  times  at  which 


PROB.  XXVIII.  TO  FIND  THE  NUMB.  OF  ECLIPSES  IN  A  YEAR. 


313 


eclipses  of  the  moon  may  or  must  occur,  after  the  same  manner 
as  for  eclipses  of  the  sun,  only  making  use  of  the  limits  35  and 
25,  instead  of  53  and  37.* 

Note.  An  eclipse  of  the  moon  will  be  visible  at  a  given  place, 
if  the  time  of  the  eclipse  thus  found  nearly  and  reduced  to  the 
meridian  of  the  place  comes  in  the  night. 

Exam.  1.  Required  the  eclipses  that  may  be  expected  in  the 
year  1840,  and  the  times  nearly  at  which  they  will  take  place. 


For  the  Eclipses  of  the  Sun. 


M.New  Muon. 

I. 

II. 

III. 

IV. 

N. 

d.    h.  m. 

1840, 

3  10  30 

0085 

6386 

65 

63 

844 

2  lun. 

59     1  28 

1617 

1434 

31 

98 

170 

62  11  58 

1702 

7820 

96 

61 

014 

60 

As  the  sum  of  the  numbers 

March 

2  11  58 

I. 

8     3 

N  comes  within  37  of"  0,  there 

11. 

19  38 

must  be  an  eclipse. 

Ill 

12 

IV. 

13 

March 

3  16     4 

Mean 

time  at 

Greei 

iwich 

I 

1 

M.New  Moon. 

I. 

II. 

III.  1  IV. 

N. 

d. 

h.  m. 

1840, 

3 

10  30 

0085 

6386 

65 

63 

844 

8  lun. 

236 

5  52 

6468 

5737 

22 

93 

C82 

239 

16  22 

6553 

2123 

87 

56 

526 

213 

As  the  sum  of  the  numbers 

Aug. 

26 

16  22 

I. 

0  54 

N    comes  within    37  of  500, 

II. 

0  49 

there  must  be  an  eclipse. 

III. 

15 

IV. 

16 

Aug. 

26 

18  36 

Mean 

time  at 

Greer 

wich 

*  The  numbers  53,  37,  and  35,  25,  are  the  lunar  and  solar  ecliptic  limits,  a? 
determined  by  Delambre.  The  limits  given  in  the  text,  converted  into  thou- 
sandth  parts  of  the  circle,  are  55,  37,  and  37,  21.  , 

40 


314 


ASTRONOMY. 


For  the  Eclipses  of  the  Moon. 


1840, 
i  lun. 

M.  Full  Moon. 

I. 

II. 

111. 

IV. 

N. 

(1.   h.   ni. 

3   10  311 

14  18  22 

0085 

404 

f;3<6 

5359 

65 

58 

63 
50 

844 
43 

1  lun. 

18     4  52 
29  12  44 

489 

808 

1745 
717 

23 
15 

38 

13 
99 

887 
85 

47  17  36 
31 

1297 

2462 

12  '  972 

Febr. 

I. 

II. 

III. 

IV. 

16  17  36 

7  27 

0  23 

5 

27 

As  the  sum  of  the  numbers 
N,   althougli  it  comes  within 
35    of   1000,    does    not    come 
within  25,  the  eclipse  may  be 
considered   doubtful. 

Febr. 

17     1  58 

Mean  time  at  Greenwich 

1 
1 

M.  Full  Moon. 

I. 

II. 

III. 

IV. 

N. 

d.  h.  m. 

1840, 

18     4  52 

4R9 

1745 

23 

13 

887 

7  lun. 

206  17     8 

5659 

5020 

7 

94 

596 

224  22     0 

6148 

6765 

30 

07 

483 

213 

As  tiie  sum  of  tlie  numbers 

Auff. 

11  22     0 

I. 

1  37 

N    comes   within  25   of   500, 

II. 

19  16 

there  must  be  an  eclipse. 

III. 

3 

IV. 

25 

Aug. 

12  19  21 

Mean 

time  at 

Greer 

wich 

2.  Required  the  eclipses  that  may  be  expected  in  the  year  1839, 
and  the  times  nearly  at  which  they  will  take  place,  expressed  in 
mean  civil  time  at  New  York. 

Ans.  One  of  the  sun  on  the  15th  of  March  at  9h.  20m.  A.  M. ; 
and  one  of  the  sun  on  the  7th  of  September  at  5h.  24m.  P.  M. 

3.  Required  the  eclipses  that  may  be  expected  in  the  year  1841, 
and  the  times  nearly  at  which  they  will  take  place,  expressed  in 
mean  civil  time  at  New  York. 

Ans.  Four  of  the  sun,  namely,  one  on  the  22nd  of  January  at 
12h.  18m.  P.  M. ;  one  on  the  21st  of  February  at  6h.  17m.  A.  M. ; 
one  on  the  18th  of  July  at  9h.  24m.  A.  M. ;  and  one  on  the  16th 


PROB.    XXIX.    TO    CALCULATE    A    LUNAR    ECLIPSE.  315 

of  August  at  4h.  28m.  P.  M. :  and  two  of  the  moon,  namely,  one 
on  the  5th  of  February  at  9h.  10m.  P.  M. ;  and  one  on  the  2d  of 
August  at  5h.  5m.  A.  M. 

The  ecUpses  of  the  sun  in  January  and  August  may  be  consid- 
ered as  doubtful. 


PROBLEM   XXIX. 

To  calculate  an  Eclipse  of  the  Moon. 
The  calculation  of  the  circumstances  of  a  lunar  ec'ipse  is  effect- 
ed with  the  following  fundamental  data,  derived  from  the  tables 
of  the  sun  and  moon  : 

Approximate  Time  of  Full  Moon  (at  Greenwich)        T 

Sun's  Longitude  at  that  time,  .         _         .  L 

Do.  Hourly  Motion,  _         .         _         .  ^ 

Do.  Semi-diameter, S 

Do.  Parallax,       - p 

Moon's  Longitude,         .         .         .         .         _  / 

Do.      Latitude,  _         _         _         .         .  x 

Do.      Equatorial  Parallax,  .         .         .  p 

Do.      Semi-diameter,  _         -        .         _  d 

Do.      Hourly  Motion  in  longitude,       -        -  m 

Do.      Hourly  Motion  in  latitude,  -         -  n 

We  obtain  the  time  T  by  Problem  XXVH  ;  the  quantities  apper- 
taining to  the  sun,  namely,  L,  s,  and  5,  by  Problem  IX  ;*  and  those 
which  have  relation  to  the  moon,  namely,  Z,  X,  P,  d,  m,  and  n,  by 
Problem  XIV. 

From  these  quantities  we  derive  the  following : 

True  Time  of  Full  Moon,  (at  given  place)         -         T' 
Moon's  Latitude  at  that  time,  -         -         -         X' 

Semi-diameter  of  earth's  shadow,     -         -         -         S 
Inclination  of  Moon's  relative  orbit,  -         -        I 

T  being  known,  T'  is  found  as  explained  in  Problem  XXVII. 
To  obtain  X',  we  state  the  following  proportion, 

1  hour :  correction  for  the  time  of  full  moon  :  :  n  :  x; 

•  p  may  be  taken  =  9". 


316  ASTRONOMY. 

from  this  we  deduce  the  value  of  x  ;  and  thence  find  X'  by  the 
equation, 

X'  =  X  ±  ar. 

When  the  true  time  of  full  moon,  expressed  in  mean  time  at 
Greenwich,  is  later  than  the  approximate  time,  the  wpyer  sign  is 
to  be  used,  if  the  latitude  is  increasing,  the  lower  if  it  is  decreas- 
ing ;  but  when  the  true  time  is  earlier  than  the  approximate  time, 
the  lower  sign  is  to  be  used  if  the  latitude  is  increasing ;  the  upper, 
if  it  is  decreasing. 

The  value  of  S  is  derived  from  the  equation, 

S  =  (P+p-6)  +  ^V(P+p-<5); 

and  the  angle  I  from  the  formula, 

log.  tang  I  =  log.  n  +  ar.  co.  log.  [m  —  s). 

The  foregoing  quantities  having  all  been  determined,  the  va- 
rious circumstances  of  the  eclipse  may  be  calculated  by  the  follow- 
ing formulae  : 

For  the  Time  of  the  Middle  of  the  Eclipse. 

3.55630  +  log.  COS.  I  +  ar.  co.  log.  (m  —  5)  —  10  =  R; 
log.  ^  =  R  +  log.  X'  +  log.  sin  I  —  10 ; 
M  =  T'  ±  ^ : 
t  =  interval  between  time  of  middle  of  eclipse  and  time  of  full 
moon  ;  M  =  time  of  middle  of  the  eclipse. 

The  upper  sign  is  to  be  taken  in  the  last  equation  when  the 
latitude  is  decreasing  ;  the  lower,  when  it  is  increasing. 

For  the  Times  of  Beginning  and  End. 

log.  c  =  log.  X'  -f  log.  cos.  I  —  10 ; 

log.  V  =  log-(S  +  ^  +  ^)  +  ^og-(S+<^-g)  +  R  ; 

B  =  M  —  v,  and  E  =  M  +  V  : 
V  =  half  duration  of  the  eclipse ;   B  =  time  of  beginning ;    and 
E  =  time  of  end. 

Note.  If  c  is  equal  to  or  greater  than  S  +  0?,  there  cannot  be 
an  eclipse. 


PROB.  XXIX.    TO  CALCULATE  A  LUNAR  ECLIPSE.  317 

For  the  Times  of  Beginning  and  End  of  the  Total  Eclipse. 
l,^,.,_log:-(S-^  +  c)  +  log.(S-(^-c)     p 

B'  =  M  — t;',  andE'  =  M  +  ?^'; 
v'  =  half  duration  of  the  toial  eclipse  ;  B'  =  time  of  beginning  of 
total  eclipse  ;  and  E'  =  time  of  end  of  total  eclipse. 

Note.  When  c  is  greater  ihan  S  —  d.,  the  eclipse  cannot  be 
total. 

For  the  Q,uantity  of  the  Eclipse. 
log.  a  =  0.77815  +  bg.  (S  +  ^  ~  c)  +  ar.  co.  log.  (^  —  10 ; 
Q,  =  the  quantity  of  the  eclipse  in  digits. 

Note  1.  An  eclipse  of  the  moon  begins  on  the  eastern  limb,  and 
ends  on  the  western.  In  partial  eclipses  the  southern  part  of  the 
moon  is  eclipsed  when  the  latitude  is  north,  and  the  northern 
part  when  the  latitude  is  south. 

Note  2.  When  the  eclipse  commences  before  sunset,  and  ends 
after  sunset,  the  moon  will  rise  more  or  less  eclipsed.  To  obtain 
the  quantity  of  the  eclipse  at  the  time  of  the  moon's  rising,  find 
the  moon's  hourly  motion  on  the  relative  orbit  by  the  equation, 

log.  h  =  log.  (m  —  s)  -\-  ar.  co.  log.  cos.  I ; 
in  which  h  —  hourly  motion  or  relative  orbit.     Also  find  the  in- 
terval between  the  time  of  sunset  and  the  time  of  the  middle  of 
the  eclipse,  which  call  i.     Then, 

1  hour  :  i  :  '.  h  :  X. 
Deduce  the  value  of  x  from  this  proportion,  and  substitute  it 
in  the  equation, 

in  which  c  designates  the  same  quantity  as  in  previous  formulae. 
Find  the  value  of  c',  and  use  it  in  place  of  c  in  the  above  formula 
for  the  quantity  of  the  eclipse,  and  it  will  give  the  quantity  of 
the  eclipse  at  the  time  of  the  moon's  rising.  When  the  eclipse 
begins  before  and  ends  after  sunrise,  the  quantity  of  the  eclipse 
at  the  time  of  the  moon's  setting  may  be  found  in  the  same  man- 
ner, only  using  sunrise  instead  of  sunset. 

Example.  Required  to  calculate,  for  the  meridian  of  New- York, 
the  eclipse  of  the  moon  in  October,  1837. 


318  ASTRONOMY. 

Elements. 

Approximate  time  of  full  moon,         -  T  =  11^-  lO""-  (Oct.  13) 

Sun's  longitude  at  that  time,      -        -  L  =  G''-  20°  24'  28" 

Do.  hourly  motion,         -        -         -  s  =  2  29 

Do.  semi- diameter,          -        -        -  5  =  16     4 

Do.  parallax,          -        -        -        -  -p  —  9 

Moon's  longitude,     -        -        -        -  Z  =  0     20   21  51 

Do.  latitude,           -        -        -        -  X  =  11  28 

Do.  equatorial  parallax,  -         -         -  P  =  59  32 

Do.  semi-diameter,          -        -        -  d  =  1613 

Do.  hourly  motion  in  long.     -         -  m=  35  54 

Do.  hourly  motion  in  lat. (tending  north),  n  =  3  19 

Approx.  time  of  full  moon,  October,         -         13'5-  11^-  10™- 00^- 
Correction  found  by  Prob.  XXVII,  -  +  4     42 

True  time,  in  mean  time  at  Greenwich,    -         13     11     14     42 
Diff.  of  meridians,  .         .         .         .  4     56       4 

True  time, in  mean  time  at  New  York,  T'=     13       6     18     38 

60™-  :  4""-  42^-  :  :  3'  19"  :  x  =  16". 
Moon's  lat.  at  approx.  time,        -        -        -        X  =   11'   28"  S. 
Correction, x  =     — 16 


Moon's  lat.  at  true  time,     -        -        -        -        X'=   11    12 

Moon's  equatorial  parallax,        -        -        -        -        P  =  59'  32" 
Sun's  do. p  =         9 

Sum, 59  41 

Sun's  semi-diameter, 5=164 

Diff.         -        -        -        -        -        -        V  -{-p—  S  =A3  37 

Add _V(P+p_<5)=       44 

Semi-diameter  of  earth's  shadow,        -        -        -        S  =  44  21 

Moon's  hor.  mot.  less  sun's  (m  —  s)  =  2005"-  Ar.  co.  log.  6.69789 
Moon's  hor.  motion  in  latitude,     n  =    199  -       log.  2.29885 

Liclinationofrel.  orbit,  I  =  5°40'     -        -        -       tan.  8.99674 


I    - 

m  —  y 


PROB.  XXIX.    TO  CALCULATE  JL  LUNAR  ECLIPSE.  319 

Time  of  Middle. 

3.5.5630 

-      5"  40'        -        -         COS.  9.99787 

2005"  Ar.  CO.  log.  6.69789 


X      - 
I      - 

t      - 
T'    - 

Middle, 


672" 


5°  40' 


Oh.  im.  58..  ^  1x3s 
6  18    38  P.M. 


6  20    36  P.M. 


R.  0.25206 
log.  2.82737 
sill.  8.99450 


log.  2.07393 


Times  of  Beginning  and  End. 


X' 

- 

. 

log.  2.82737 

I 

- 

11' 9"  =  669"    - 

COS.  9.99787 

c          -       - 

log.  2.82524 

S+rf+c     - 

- 

. 

-    4303"    - 

log.  3.63377 

S-\-d  —  c    - 

-    2965      - 

log.  3.47202 

2  )  7.10579 

3.55289 

Ih. 

46™ 

•  22^-  =  6382«-     - 

R.    0.25206 

V              -           - 

log.  3.80495 

Middle, 

6 

20 

36 

Beginning,   - 

4 

34 

14  P.  M. 

End,    - 

8 

6 

58  P.M. 

S  —  d-\-c 

- 

. 

2357"     - 

log.  3.37236 

S  —  d  —  c 

1019       - 

log.  3.00817 

2)  6.38053 

3.19026 
R.    0.25206 

O'l.  46m.  9s.  =2769^- 


log.  3.44232 


d^U 

ASTRONOMY. 

Middle, 

Oh.  46m.  gs.  ^  2769^-     -        log.  3.44232 
6    20  36 

Beg.  of  total  eclipse, 
End  of  total  eclipse, 

5    34  27  P.  M. 
7      6  45  P.M. 

S  +  d  —  c     - 

d            -        -        - 

0.77815 

log.  3.47202 

-     973"  Ar.  CO.  log.  7.01189 

Quantity, 


18.3  digits, 


log.  1.26206 


PROBLEM   XXX. 


To  calculate  an  Eclipse  of  the  Sun,  for  a  given  Place. 

Having  found  by  the  rule  given  in  the  note  to  Problem 
XXVIII,  that  there  is  a  probability  that  the  eclipse  will  be  visi- 
ble at  the  given  place,  and  calculated  the  approximate  time  of  new 
moon  by  Problem  XXVII,  find  from  the  tables  for  this  time  or 
for  the  nearest  whole  or  half  hour,  the  sun's  longitude,  hourly 
motion,  and  semi-diameter  ;  and  the  moon's  longitude,  latitude, 
equatorial  parallax,  semi-diameter,  and  hourly  motions  in  longi- 
tude and  latitude.  Find  also  by  Problem  XVI,  the  longitude 
and  altitude  of  the  nonagesimal  degree  ;  and  thence  compute  by 
Problem  XVII,  the  apparent  longitude,  latitude,  and  augmented 
semi-diameter  of  the  moon,  (using  the  relative  horizontal  paral- 
lax). With  these  data  compute  the  apparent  distance  of  the 
centres  of  the  sun  and  moon,  at  the  time  in  question,  by  means 
of  the  following  formulae  : 

log.  tang  d  —  log.  X'  -J-  ar.  co.  log.  a  ; 

log.  A  =  log.  a  +  ar.  CO.  log.  cos.  6 ; 
in  which, 

A  =  appar.  distance  of  centres  ; 
X'  =  appar.  Lat.  of  Moon  ; 

a  -  Difi".  of  appar.  Long,  of  Moon  and  Sun  =  diff.  of  appar. 
long,  of  Moon  (found  as  above)  and  true  long,  of  Sun. 


PROB.  XXX.    TO    CALCULATE  A  SOLAR  ECLIPSE.  321 

6  is  an  auxiliary  arc.  The  value  of  d  being  derived  from  the 
first  equation,  the  second  will  then  make  known  the  value  o(  a, 

a  and  X'  are  in  every  instance  to  be  effected  with  the  positive 
sign.* 

For  the  Approximate  Times  of  Beginning,  Greatest 
Obscuration,  and  End 

Let  the  time  for  which  the  above  calculations  are  made,, 
be  denoted  by  T.  If  the  apparent  distance  of  the  centres 
of  the  sun  and  moon,  found  for  the  time  T,  is  less  than  the 
sum  of  their  apparent  semi-diameters,  there  is  an  eclipse  at  this 
time.  But  if  it  is  greater,  either  the  eclipse  has  not  yet  com- 
menced, or  it  has  already  terminated.  It  has  not  commenced 
if  the  apparent  longitude  of  the  moon  is  less  than  the  longitude 
of  the  sun  :  and  has  terminated,  if  the  apparent  longitude  of  the 
moon  is  greater  than  the  longitude  of  the  sun. 

1.  If  there  should  be  an  eclipse  at  the  time  T,  from  the  sun's 
longitude  and  hourly  motion  in  longitude,  and  the  moon's  lonafi- 
tude  and  latitude,  and  hourly  motions  in  longitude  and  latitude, 
found  for  this  time,  calculate  the  longitudes  and  the  moon's  lati- 
tude for  two  instants  respectively  an  hour  before,  and  an  hour 
after  the  time  T.  The  semi -diameter  of  the  sun,  and  the  equa- 
torial parallax  and  semi-diameter  of  the  moon,  may,  in  our  pre- 
sent inquiry,  be  regarded  as  remaining  the  same  during  the 
eclipse.  Find  the  apparent  longitude  and  latitude,  and  the  aug- 
mented semi-diameter  of  the  moon  (using  in  all  cases  the  relative 
parallax),  and  thence  compute  by  the  formulas  already  given,  the 
apparent  distance  of  the  centres  of  the  sun  and  moon  at  the  two 
instants  in  question. 

Observe  for  each  result,  whether  it  is  less  or  greater  than  the 
sum  of  the  apparent  semi-diameters  of  the  two  bodies.     If  the 


*  The  apparent  distance  or  tlie  centres  A  may  be  found  without  the  aid  of  loga- 
rithms by  means  of  the  follow. ng  equation  : 


A  =  v/  r/^  +  y-K 
If  the  logarithmic  formulEe  are  used,  it  will  be  sufficient  here  to  take  out  the  angle 
6  to  the  nearest  minute.     When  we  liave  occjsion  to  obtain  the  distancs  of  tho 
centres  exact  to  within  a  small  fraction  of  a  second,  d  must  bo  taken  to  the  nearest 
tens  of  seconds,  if  it  exceeds  20°  or  30°. 

41 


322  ASTRONOMY. 

moon  is  apparently  on  the  same  side  of  the  sun  at  the  times  T 
and  T  -(-  Ih.,  take  the  difference  of  the  distances  of  the  two  bo- 
dies in  apparent  longitude  at  these  times,  but  if  it  is  on  opposite 
sides,  take  their  sum,  and  it  will  be  the  variation  of  this  distance 
in  the  hour  following  T.  Find  in  like  manner  tlie  variation  of 
the  distance  during  the  hour  preceding  T.  Then,  if  the  apparent 
distance  of  the  centres  at  the  times  (T  —  Ih.),  (T  +  Ih.)  is  less 
than  the  sum  of  the  apparent  semi-diameters,  deduce  from  these 
results  the  variations  of  the  distance  in  apparent  longitude  during 
the  preceding  and  following  hours,  allowing  for  the  second  dif- 
ference, and  observing  whether  the  two  bodies  are  approaching 
each  other  or  receding  from  each  other.  Thence,  find  the  dis- 
tance in  apparent  longitude  at  the  times  (T  —  2h.),  (T  +  2h.) 
Find  by  the  same  method  the  apparent  latitude  of  the  moon  at 
the  instants  (T  —  2h.),  (T  +  2h.),  observing  that  the  variation 
of  the  apparent  latitude  in  any  given  interval  is  the  difference 
between  the  latitudes  at  the  beginning  and  end  of  it,  if  they  are 
both  of  the  same  name  ;  their  sum,  if  they  are  of  opposite  names. 

From  these  results  derive  the  apparent  distance  of  the  centres 
of  the  sun  and  moon  at  the  two  instants  in  question. 

If  there  should  still  be  an  eclipse  at  the  time  (T  +  2h.)  or 
(T  —  2h.),  find  by  the  same  method  the  distance  of  the  centres 
at  the  time  (T  +  3h.)  or  (T  —  3h.)  These  calculations  being 
effected,  the  times  of  the  beginning,  greatest  obscuration,  and  end 
of  the  eclipse  will  fall  between  some  of  the  instants  T,  (T  — 
Ih.),  (T  +  Ih.)  &.C.,  for  which  the  apparent  distance  of  the  cen 
tres  is  computed. 

2.  If  the  eclipse  occurs  after  the  time  T,  the  different  phases 
will  happen  between  the  instants  T,  (T  +  Ih.),  (T  -f  2h.),  <fec. 
Find  the  apparent  distance  of  the  centres  of  the  sun  and  moon 
for  the  times  (T  -f-  Ih.),  (T  +  2h.),  by  the  same  method  as  that 
by  which  it  is  found  for  the  times  (T  +  Ih.),  (T  —  Ih.),  in  the 
case  just  considered.  Then,  if  the  eclipse  has  not  terminated, 
deduce  the  distance  of  the  moon  from  the  sun  in  apparent  longi- 
tude, and  the  moon's  apparent  latitude,  for  the  time  (T  +  3h.), 
from  these  distances  and  latitudes  at  the  times  T,  (T  +  Ih.), 
(T  -f-  2h.)  ;  as  in  the  preceding  case  the  distance  and  latitude  for 
the  time  (T  +  2h.)  were  deduced  from  the  same  at  the  times 
(T  ~  Ih.),  T,  (T  +  Ih.).     With  the  results  obtained  compute 


PROS.  XXX,     TO  CALCULATE  A  SOLAR  ECLIPSE.  323 

the  apparent  distance  of  the  centres  of  the  two  bodies  at  the 
time  (T  +  3h.) 

3.  In  case  the  eclipse  occurs  before  the  time  T,  the  apparent 
distance  of  the  centres  mast  be  found  by  similar  methods  for  the 
times  (T  —  Ih.),  (T  —  2h.),  &c. 

The  calculation  is  to  be  continued  until  the  distance,  from 
being  less,  becomes  greater  than  the  sum  of  the  semi-diameters. 

Now,  let  h  —  variation  of  apparent  distance  of  centres  in  the 
interval  of  one  hour  comprised  between  the  first  two  of  the  in- 
stants for  which  the  distance  is  computed  ;  d  =  difference  be- 
tween the  sum  of  the  semi-diameters  of  the  sun  and  moon  and 
the  apparent  distance  of  their  centres  at  the  first  instant ;  and  t 
—  interval  between  first  instant  and  the  time  of  the  beginning:  of 
the  eclipse.     Then, 

h:d:\  60™-  :  t  (nearly). 
Find  the  value  of  ^  given  by  this  proportion,  and  add  it  to  the 
time  at  the  first  instant,  and  the  result  will  be  a  first  approxima- 
tion to  the  time  of  the  beginning  of  the  eclipse,  which  call  h. 
Find,  by  interpolation,*  the  distance  of  the  moon  from  the  sun  in 
apparent  longitude  (a),  and  the  moon's  apparent  latitude  ( •.'),  for 
this  time,  and  thence  compute  the  apparent  distance  of  the  cen- 
tres. Take  h  —  variation  of  apparent  distance  in  the  interval 
between  the  time  h  and  the  nearest  of  the  two  instants  above 
mentioned,  between  which  the  beginning  falls,  and  d  =  difference 
between  the  apparent  distance  of  the  centres  at  the  time  b  and  the 
sum  of  the  semi-diameters,  and  compute  again  the  value  of  t.    Add 


*  The  second  differences  may  easily  be  taken  into  the  account  in  finding  the 
quantities  a  and  X'  for  the  time  b.  Thus,  let  k  =  variation  of  a  for  the  interval  of 
an  hour  comprised  between  the  instants  above  mentioned,  k'  =  same  for  the 
succseding  hour,  and  i  =  interval  between  b  and  the  nearer  of  the  two  instants, 

k  k k' 

(in  minutes).     Then,  if  we  put  /  ==  _,  c  = — ,  and  v  =  var.  of  a  in  inter- 

6  36 

val  i. 


.■|/±(«  +  |) 


10 

The  uppor  sign  is  to  be  used  when  the  time  b  is  nearer  the  first  than  the  second 
instant,  the  lower  wlien  it  is  nearer  the  second  than  the  first.  The  error  by  tliis 
method  will  not  exceed  the  number  c  (supposing  the  changes  of  k,  k'  from  10m. 
to  10m.  to  increase  or  decrease  by  equal  degrees'). 


324  ASTRONOMY. 

this  to  the  time  b,  or  subtract  it  from  it,  according  as  b  is  before 
or  after  the  beginning,  and  the  result  will  be  a  second  approxima- 
tion to  the  time  of  the  beginning,  which  call  B.  If  necessary,  a 
result  still  more  approximate  may  be  had,  by  taking  h  =  variation 
of  apparent  distance  of  centres  in  the  interval  B  —  b.d=  differ- 
ence between  apparent  distance  at  the  time  B  and  sum  of  semi- 
diameters,  finding  anew  the  value  of  t  given  by  the  preceding 
proportion,  and  adding  it  to  or  subtracting  it  from,  as  the  case  may 
be,  the  time  B. 

The  end  of  the  eclipse  will  fall  between  the  last  two  of  the 
several  instants  for  which  the  apparent  distance  of  the  centres 
of  the  moon  and  sun  have  been  computed.  The  approximate 
time  of  the  end  is  found  by  the  same  method  as  that  of  the  be- 
ginning.* 

The  middle  of  the  interval  between  the  approximate  times  of 
the  beginninof  and  end  of  the  eclipse,  will  be  a  first  approximation 
to  the  time  of  greatest  obscuration. 

Note.  When  the  object  is  merely  to  prepare  for  an  observation, 
results  sufliciently  near  the  truth  may  be  obtained  by  a  graphical 
construction.  The  elements  of  the  construction  are  the  difference 
of  the  apparent  longitudes  of  the  moon  and  sun,  and  the  apparent 
latitude  of  the  moon,  found  as  above,  for  two  or  more  instants  dur- 
ing the  continuance  of  the  eclipse.  Draw  a  right  line  E  F  (Fig.  78), 
to  represent  the  ecliptic,  assume  on  it  some  point  C  for  the  position 
of  the  sun  at  the  instant  of  apparent  conjunction,  and  lay  off  C  A, 
C  A',  equal  to  the  two  differences  of  apparent  longitude  ;  and  to  the 
right  or  left,  according  as  the  moon  is  to  the  west  or  east  of  the  sun 
at  the  instants  for  which  the  calculations  have  been  made.  Erect 
the  perpendiculars  A  p,  A'  p',  and  mark  off  A  a,  A'  a'  equal  to 
the  two  apparent  latitudes.  Tlirough  a,  a',  draw  aright  line,  and 
it  will  be  the  apparent  relative  orbit  of  the  moon,  or  v/ill  differ  but 
little  from  it.  From  C  let  fall  the  perpendicular  C  m  upon  the  rela- 
tive orbit,  771  will  be  the  apparent  place  of  the  moon  at  the  instant 
of  greatest  obscuration.     Take  a  distance  in  the  dividers  equal  to 


*  In  effecting  the  reductions  of  tlic  qiiantitiss  a  and  A'  to  the  first  approximate 
time  of  end,  k'  must  stand  for  tlie  variation  of  a  during  the  hour  preceding  that 
comprised  between  the  last  two  instants,  and  the  last  instant  must  be  substituted 
for  the  first.    (See  Note,  p.  323.) 


PROB.  XXX.    TO  CALCULATE  A  SOLAR  ECLIPSE,  325 

the  sum  of  the  apparent  semi-diameters  of  the  moon  and  sun,  and 
placing  one  foot  of  it  at  C,  mark  off  with  the  other  the  points/,/', 
for  tlie  beginning  and  end  of  the  eclipse,  and  by  means  of  a  square 
mark  on  E  F  the  points  b,  e,  which  answer  to  the  beginning  and  end. 
If  the  eclipse  be  total  or  annular,  mark  the  points  of  immersion  and 
emersion,  g,  g',  with  an  opening  in  the  dividers  equal  to  the  differ- 
ence of  the  semi-diameters,  and  find  the  corresponding  points  b',  e' 
on  the  line  E  F. 

If  the  calculations  are  made  from  hour  to  hour,  the  distance  A  A' 
is  the  apparent  relative  hourly  motion  of  the  sun  and  moon  in  long- 
itude. This  distance  laid  off  repeatedly  to  the  right  and  left,  will 
determine  the  points  1, 2,  (fcc,  answering  to  Ih.,  2h.,  (fee.  before  and 
after  the  times  for  which  the  calculations  are  made.  If  the  spaces 
in  which  the  points  b,  e,  answering  to  the  beginning  and  end  of  the 
eclipse,  occur,  be  divided  into  quarters,  and  then  sub-divided  into 
three  equal  parts  or  five  minute  spaces,  the  appoximate  times  of  the 
beginning  and  end  of  the  eclipse  will  become  known. 

From  the  point  m,  as  a  centre,  describe  the  lunar  disc  ;  and 
from  the  point  C,  as  a  centre,  describe  the  sun's  disc,  and  we  shall 
have  the  figure  of  the  greatest  eclipse.  The  quantity  of  the 
eclipse  will  result  from  the  proportion, 

S  N  :  M  N  :  :  12  :  number  of  digits  eclipsed. 

Draw  from  the  centre  C  to  the  place  of  commencement/  the 
line  C/;  and  through  the  same  point  C  raise  a  perpendicular  to 
the  ecliptic.  With  the  declination  and  lono-itude  of  the  sun  at  the 
time  of  the  beginning,  calculate  its  angle  of  position  by  Problem 
XIII,  and  lay  it  off  in  the  figure,  placing  the  circle  of  declina- 
tion C  P  to  the  left  if  the  tangent  of  the  angle  of  position  be  posi- 
tive, to  the  right  if  it  be  negative. 

Compute  also  for  the  time  of  beginning,  the  angle  of  the 
vertical  of  the  sun  with  the  circle  of  declination,  that  is,  the  angle 
PSZ  in  Figure  18,  for  which  we  have  in  the  triangle  PSZ 
the  side  P  S  =  co-declination,  the  side  P  Z  =  co-latitude,  and 
the  included  angle  Z  P  S.  (The  requisite  formulae  are  given  in  the 
Appendix).  Form  this  angle  in  the  figure  at  the  point  C,  plac- 
ing C  Z  to  the  right  or  left  of  C  P,  according  as  the  time  is  in 
the  forenoon  or  afternoon,  C  Z  will  be  the  vertical,  and  Z  the 
vertex,  or  highest  point  of  the  sun.     The  arc  Z  /  on  the  limb  of 


326  ASTRONOMY. 

the  sun,  will  be  the  angular  distance  from  the  vertex  of  the  point 
on  the  limb  at  which  the  eclipse  commences. 

For  the  True  Times  of  Beginning,  Greatest  Obscuration, 
and  End. 

The  approximate  times  of  beginnino-.  greatest  obscuration,  and 
end  of  the  eclipse,  being  calculated  by  the  rules  which  have  been 
given,  find  from  the  tables,  the  moon's  longitude,  latitude,  equato- 
rial parallax,  semi-diameter,  and  hourly  motions  in  longitude  and 
latitude,  for  the  approximate  time  of  greatest  obscuration.  "With 
the  moon's  longitude  and  latitude,  and  lioarly  motions  in  longi- 
tude and  latitude,  found  for  this  time,  calculate  the  longitude  and 
latitude  for  the  approximate  times  of  beginning  and  end.  The 
parallax  and  semi-diameter  may,  without  material  error,  be  con- 
sidered the  same  during  the  eclipse.  With  the  moon's  true  longi- 
tude, latitude  and  semi-diameter  at  the  approximate  times  of  begin- 
ning, greatest  obscuration,  and  end,  calculate  its  apparent  longi- 
tude and  latitude,  and  augmented  semi-diameter,  for  these  several 
times,  (making  use  of  the  relative  parallax).  With  the  sun's 
longitude  and  hourly  motion  previously  found  for  the  approxi- 
mate time  of  new  moon,  find  his  longitude  at  the  approximate 
times  of  beginning,  greatest  obscuration,  and  end.  The  sun's 
semi-diameter  found  for  the  approximate  time  of  new  moon  will 
serve  also  for  any  time  during  the  eclipse.  With  the  data  thus 
obtained,  calculate  by  the  formulae  giv^en  on  page  320,  the  appa- 
rent distance  of  the  centres  of  the  sun  and  moon  at  the  approxi- 
mate times  of  the  three  phases. 

Note.  When  very  great  accuracy  is  required,  the  moon's  lon- 
gitude, latitude,  equatorial  parallax,  semi-diameter,  and  hourly 
motions  in  longitude  and  latitude,  must  be  calculated  directly 
from  the  tables,  for  the  approximate  times  of  the  beginning  and 
end,  as  well  as  for  that  of  the  greatest  obscuration. 

For  the  Beginning. 
Subtract  the  apparent  longitude  of  the  moon  at  the  ap- 
proximate time  of  beginnino^  from  the  true  longitude  of 
the  sun  at  the  same  time,  and  denote  the  difference  by  a.  Do 
the  same  for  the  approximate  time  of  greatest  obscuration. 
Subtract  the  latter  result  from  the  former,  paying  attention  to 


PROB.  XXX.    TO  CALCULATE  A  SOLAR  ECLIPSE.       327 

the  signs,  and  call  the  remainder  k.  Next,  take  the  difference 
between  the  apparent  latitudes  of  the  moon  at  the  approximate 
times  of  beginnins:  and  greatest  obscuration,  if  they  are  of  the  same 
name  ;  their  sum,  if  they  are  of  opposite  names  ;  and  denote  the 
diiference  or  sum,  as  the  case  may  be,  by  71.  This  done,  com- 
pute the  correction  to  be  applied  to  the  approximate  time  of  be- 
ginning, by  means  of  the  following  formulae  : 

log.  b  =  log.  a  +  log.  k  +  ar.  co.  log.  n  —  10 ; 

c  =  X'  —  6,   S  =  c/  -f  6  —  5"  ; 

log.  t  =  log.  (S  +  A)  +  log.  (S  —  A)  +  ar.  co.  log.  n  +  nr. 
CO.'  log.  c  +  log.  L  +  1.47712  —  20  ; 

in  which, 

t  =  Correction  of  approx.  time  of  beginn.  (required) ; 

a  =  Diff.  of  appar.  long,  of  Moon  and  Sun  at  approx.  time  ; 

L  =  Half  duration   of  eclipse  in   minutes   (known  approx- 
imately) ; 

k  =  Appar.  relative  motion  of  Sun  and  Moon  in  long,  in  the 
interval  L  ; 

n  =  Moon's  appar.  motion  in  lat.  in  same  interval ; 

X'=  Moon's  appar.  lat. ; 

d  =  Augmented  semi-diameter  of  Moon  ; 

6  =  Semi-diam.  of  Sun  ; 

A  =  Appar.  distance  of  centres  of  Sun  and  Moon. 

b  and  c  are  auxiliary  quantities. 

First  find  the  value  of  b  by  the  first  equation,  and  substitute  it 
in  the  second.  Then  derive  the  values  of  c  and  S  from  the 
second  and  third  equations,  and  substitute  them  in  the  fourth, 
and  it  will  make  known  the  value  of  ^,  which  is  to  be  applied  to 
the  approximate  time  of  the  beginning  of  the  eclipse  according 
to  its  sign. 

The  quantities  a,  A:,  n,  <fcc.  are  all  to  be  expressed  in  seconds. 
The  apparent  latitude  X'  must  be  affected  with  the  negative 
sign,  when  it  is  south.  The  motion  in  latitude,  w,  must  also 
have  the  negative  sign  in  case  the  moon  is  apparently  receding 
from  the  north  pole,   a  and  k  are  always  positive. 


328 


ASTRONOMY. 


The  result  may  be  verified,  and  corrected,  by  computing  the 
apparent  distance  of  the  centres  at  the  time  found,  and  comparing 
it  with  the  sum  of  the  semi-diameters  minus  5". 

Note.  When  great  precision  is  desired,  the  quantities  k  and  71 
must  be  found  for  some  shorter  interval  than  the  half  duration 
of  the  eclipse.  Let  some  instant  be  fixed  upon,  some  five  or  ten 
minutes  before  or  after  the  approximate  time  of  the  beginning  of 
the  eclipse,  according  as  the  contact  takes  place  before  or  after. 
For  this  time  deduce  the  longitude  and  latitude  of  the  moon, 
from  the  longitude  and  latitude  at  the  approximate  time  of  be- 
ginning, by  means  of  their  hourly  variations  ;  and  thence  calcu- 
late the  apparent  longitude  and  latitude,  and  the  augmented 
semi-diameter.  Find  the  longitude  of  the  sun  for  the  time  in 
question,  from  its  longitude  and  hourly  motion  already  known 
for  the  approximate  time  of  beginning.  Then  proceed  according 
to  the  rule  given  above,  only  using  the  quantities  thus  found  for 
the  time  assumed,  in  place  of  the  corresponding  quantities 
answering  to  the  approximate  time  of  greatest  obscuration.  L 
will  always  represent  the  interval  for  which  k  and  n  are 
determined. 

For  the  End. 
Subtract  the  longitude  of  the  sun  at  the  approximate 
time  of  the  end  from  the  apparent  longitude  of  the  moon 
at  the  same  time.  Do  the  same  for  the  approximate  time  of 
greatest  obscuration.  Then  proceed  according  to  the  rule  for 
the  beginning,  only  substituting  every  where  the  approximate 
time  of  the  end  for  the  approximate  time  of  the  beginning,  and 
taking  in  place  of  the  formula  c  =  X'  —  6,  the  following : 

c  =  X'  +  6. 

For  the  Greatest  Obscuration. 
Take  the  sum  of  the  distances  of  the  moon  from  the  sun  in  appa- 
rent longitude  at  the  approximate  times  of  the  beginning  and  end 
of  the  eclipse,  and  call  it  k.  Take  the  difference  of  the  apparent 
latitudes  of  the  moon  at  the  same  times,  if  the  two  are  of  the  same 
name  ;  but  if  they  are  of  dififerent  names,  take  their  sum.  Denote 
the  difference  or  sum  by  n.  Let  a'  =  the  distance  of  the  moon  from 
the  sun  in  apparent  longitude  at  the  true  time  of  greatest  obscu- 


PROB.    XXX.      TO    CALCULATE    A    SOLAR    ECLIPSE.  329 

ration  ;  X'  =  the  apparent  latitude  of  the  moon  at  the  approximate 
time  of  greatest  obscuration, 

k  :  71  :  :  X' :  a'. 

Find  the  wilue  of  a'  by  this  proportion,  affecting  X',  ?/,,  /j, 
always  with  the  positive  sign. 

Ascertain  whether  the  greatest  obscuration  has  place  before  or 
after  the  apparent  conjunction,  by  observing  whether  the  appa- 
rent latitudo  of  the  moon  is  increasing  or  decreasing  about 
this  time ;  the  rule  being,  that  when  it  is  increasing,  the  great- 
est obscuration  will  occur  before  apparent  conjunction  ;  when  it 
is  decreasiitsr,  after.  If  the  approximate  and  true  times  of  great- 
est obscuration  are  both  before  or  both  after  apparent  conjunc- 
tion, from  the  value  found  for  a'  subtract  the  distance  of  the 
moon  from  the  sun  in  apparent  longitude  at  the  approximate 
time  ;  but  if  one  of  the  times  is  before  and  the  other  after  appa- 
reiit  conjunction,  take  the  sum  of  the  same  quantities.  Denote 
the  difference  or  sum  by  7n.  Also  let  D  =  duration  of  eclipse, 
and  t  -  correction  to  be  applied  to  the  approximate  time  of  o-reat- 
est  obscuration.     Then  to  find  ^,  we  have  the  proportion 

k  '.  tn  :  -.Yi  :  t. 

If  the  apparent  latitude  of  the  moon  is  decreasing,  /  is  to  be 
applied  according  to  the  sign  of  wj ;  but  if  the  apparent  latitude 
is  increasing,  it  is  to  be  applied  according  to  the  opposite  sign. 

A  still  more  exact  result  may  be  had  by  repeating  the  forego- 
ing calculations,  making  use  now  of  the  apparent  latitude  at  the 
time  just  found.  When  the  greatest  accuracy  is  required,  the 
values  of  k  and  n  may  be  found  more  exactly  after  the  same 
manner  as  for  the  beginning  or  end. 

For  the  (Quantity  of  the  Eclipse. 

Find  by  interpolation  the  apparent  latitude  of  the  moon  at  the 
true  time  of  greatest  obscuration.  With  this,  and  the  distance 
in  longitude  a  obtained  by  the  proportion  above  given,  compute 
by  ihe  formulae  on  page  320,  the  apparent  distance  of  the  cen- 
tres of  the  Sim  and  moon  at  the  time  of  greatest  obscuration. 
Subtract  this  distance  from  the  sum  of  the  apparent  semi-diam- 
eter of  the  two  bodies,  and  denote  the  remainder  by  R.     Then, 

Sun's  semi-diara.  :  R  :  :  6  digits  :  number  of  digits  eclipsed. 
42 


330 


ASTRONOMY. 


When  the  apparent  distance  of  the  centres  of  the  sun  and  moon 
at  the  time  of  greatest  obscuration  is  less  than  the  difference  be- 
tween the  sun's  semi-diameter  and  the  anirmented  semi-diameter 
of  the  moon,  the  eclipse  is  either  anmilar  or  total ;  aumilar,  when 
the  sun's  semi-diameter  is  the  greater  of  the  two ;  total,  when  it 
is  the  less. 

For  the  Beginning  and  End  of  the  Ajinular  or  Total  Eclipse. 
The  times  of  the  beginning  and  end  of  the  annular  or  total 
eclipse  may  be  found  as  follows:  the  greatest  obscuration  will 
take  place  very  nearly  at  the  middle  of  the  eclipse  in  question,  and 
will  not  differ,  at  most,  more  than  five  or  eight  minutes  (according 
as  the  eclipse  is  total  or  annular)  from  the  beginniu<?  and  end  : 
to  obtain  the  half  duration  of  the  eclipse,  and  thence  the  times 
of  the  beginning  and  end,  we  have  the  formulae 
log.  tang  d  =  log.  V  -f  ar.  co.  log.  a,  log.  k'  =  log.  k  +  ar.  co.  log.  sin  6 
S  =  5  —  d—  1",  or  S  =  rf  —  (5  +  1" ; 

loo-   c  =  log-  (S  +  A)  +  log-  (S  -  A)   . 

2  ' 

log.  t  -  ar.  CO.  log.  k'  +  log.  c  -f  log.  D  +  1.77815  —  10. 
Time  of  Begin.  =  M  —  t,  Time  of  End  =  M  +  ^, 

in  which. 

M  =  Time  of  greatest  obscuration. 

X'  =  Moon's  apparent  latitude  at  that  time. 

a   =  Distance  of  moon  from  sun  in  appar.  lone. 

k    =  Variation  of  this  distance  during  the  whole  eclipse,  or 
relative  mot.  in  appar.  long,  during  this  interval. 

k'  =  Moon's  appar.  mot.  on  relative  orbit  for  same  interval. 

6     =  Inclination  of  relative  orbit. 

8    =  Semi-diameter  of  sun. 

d   =  Augm.  semi-diam.  of  moon. 

A   =  Appar.  distance  of    centres. 

D  =  Duration  of  eclipse  (partial  and  annular  or  total). 

t    =  Half  duration  of  annular  or  total  eclipse. 

The  first  value  of  S  is  used  when  the  eclipse  is  annular,  the 
second  when  it  is  total.  The  quantities  may  all  be  regarded  as 
positive.     The  results  may  be  verified  and  corrected  by  finding 


PROB.     XXX.      TO    CALCULATE    A    SOLAR    ECLIPSE.  331 

directly  the  apparent  distance  of  the  centres  for  the  times  ob- 
tained, and  comparing  it  with  the  vakie  of  S. 

For  the  Point  of  the   Suii's  Limb  at  ivhich  the  Eclij)se 
comjnences. 

Find  the  angle  of  position  of  the  sun,  and  the  angle  between  its 
vertical  circle  and  circle  of  declination,  at  the  beginning  of  the 
eclipse,  as  explained  at  j  age  325.  Let  the  former  be  denoted  by  yj, 
and  the  latter  by  v.  Give  to  each  the  negative  sign,  if  laid  off  to- 
wards the  right ;  the  positive  sign,  if  laid  off  towards  the  left.  liCt 
a  =  distance  of  the  moon  from  the  sun  in  apparent  longitude  at 
the  beginning  of  the  eclipse  ;  X'  =  the  moon's  apparent  latitude  at 
the  same  time;  and  d  =  aiignUu-  distance  of  the  point  of  contact 
from  the  ecliptic.     Compute  the  angle  6  by  the  formula, 

log.  tang.  &  =  log.  X'  -f  ar.  co.  log.  a ; 
taking  it  always  less  than  90°,  and  positive  or  negative  according 
to  the  sign  of  its  tangent.    X'  is  negative  when  south  ;  a  is  always 
positive. 

Let  A  =  distance  on  the  limb  of  the  point  of  contact  from  the 
vertex.  The  above  operations  being  performed,  the  value  of  A 
results  from  the  equation, 

A  =  p  -f  r  -1-  90°  —  d ; 

p,  v,  and  &  being  taken  with  their  signs. 

If  the  result  is  affected  with  the  positive  sign,  the  point  first 
touched  will  lie  to  the  right  of  the  vertex.  If  with  the  negative 
sign,  it  will  lie  to  the  left  of  the  vertex. 

Note.  The  circumstances  of  an  occupation  of  a  fixed  star  by 
the  moon  may  be  calculated  in  nearly  the  same  manner  as  those 
of  a  solar  eclipse.  The  star  in  the  occultation  holds  the  place  of 
the  sun  in  the  eclipse.  The  immersion  and  emersion  of  the  star 
correspond  to  the  beginning  and  end  of  the  eclipse.  The  ele- 
ments which  ascertain  the  relative  apparent  place  and  motion  of 
the  moon  and  star,  take  the  place  of  those  which  ascertain  the 
relative  apparent  place  and  motinn  of  the  moon  and  sun.  Thus 
the  star's  longitude,  corrected  for  aberration  and  nutation  (see 
Problem  XXIII),  must  be  used  instead  of  the  sun's  longitudes; 
the  apparent  distances  of  the  moon  from  thj  s.ar  in  latitude,  in- 
stead of  the  moon's  apparent  latitudes ;  and  the  moon's  augmented 


332  ASTRONOMY. 

semi-diameter,  instead  of  the  siiin  of  tiie  -emi-diameters  of  the  sun 
and  moon.  The  difference  of  the  longitudes,  and  the  relative 
motion  in  longitude,  must  also  now  be  reduced  to  a  parallel  to 
the  ecliptic  passing  through  the  star,  fGee  Art.  4.58,  page  187). 
If  X  =  apparent  latitude  of  star,  a  -  diff.  of  appar.  longitudes  of 
moon  and  star,  and  k  =  relative  motion  in  longitude,  we  must 
substitute  in  the  formulae  for  the  eclipse,  for  X',  X'  —  X;  for  a,  a 
cos  X ;  and  for  k,  k  cos  X.  ji  will  stand  for  the  relative  motion  in 
latitude,  or  for  the  variation  of  X'  —  X. 

Example.  Required  to  calculate  an  eclipse  of  the  sun,  for  the 
latitude  and  meridian  of  New  York,  that  will  occur  on  the  18.h  of 
September,  1838. 

For  the  Approximate   Times  of  the  Phases. 
Approximate  time  of  New  Moon, 

Sept.  18'-  8'''-  49'"- 

Sun's  longitude, 175°  27'  3r'.4 

Do.  hourly  motion, 2  26  .7 

Do.  semi-diameter,  -         -         -         -       -  15  57.0 

Moon's  longitude,  .        .        .        -        175  29  19 

Do.  latitude, -  47  47 

Do.  equatorial  parallax,  -         -        -       -  53  53 

Do.  semi-diameter, 14  41 

Do.  hor.  mot.  in  long. 29  29 

Do.  hor.  mot.  in  lat.        -----  2  41 

Do.  appar.  long.  (Prob.  XVII),        -        -         175  10  26 

■  Do.  appar.  lat.V'). 2  25  N. 

Do.  augm.  semi-diameter,       ...       -  14  47 

DifF.  of  appar.  long,  (rt),  .         .         _       .  17     5 

Appar.  dist.  of  cen.  (a),  -        -        -       .  17  15 

Sum  of  semi-diameters,  _        -        .       -  30  44 

7h.  49m. 

Sun's  longitude, 175°  25'    4" 

Moon's  appar.  long.        -         -        -         -         174    47     3 

Do.  appar.  lat.  (x'), 8   12  N. 

Do.  augm.  semi-diameter,      .        -        -       -  14  49 

Diff.  of  appar.  long,  (a),  -        -         -       -  38     1         ' 


PROB.    XXX.      TO    CALCULATE    A    SOLAR    ECLIPSE. 


533 


Appar.  dist.  of  cen.  (a), 
Sum  of  semi-diameters, 


gh.  49m. 


Sun's  longitude,     - 
Moon's  appar.  long. 
Do.  appar.  lat.  (x.'), 
Do.  augm.  semi-diameter, 
Diif.  of  appar.  long.  («), 
Appar.  dist.  of  cen.  (a), 
Sum  of  semi-diameters,  - 


38' 

53" 

30 

46 

175^ 

29' 

58" 

175 

36 

15 

2 

18  S 

14 

44 

6 

17 

6 

42 

30  41 

7h  49m  2281' 

8  49   11025 

9  49  I  377 

10  49    1925 


diff.  or  k 


1256" 

1402 

1548 


492" 

N 

145 

N 

138 

S 

357 

s 

diff.  or  n 


347' 

283 
219 


diff. 


^'^^?  1298" 


sum  se  mid. 


1035 

402 

1958 


1556 


1846" 
1844 
1841 
1839 


For  the  Approximate   Time  of  Beginning. 

h  =  1298",  d  =  2333"  —  1846"  =  487" ; 

1298"  :  487"  : :  60™-  :  t  =  22'"-.5 
jh.  49m. 
22 


7h.    49m. 

Correction  for  22'"-    447 


1st  Approxi.  S^-  ll"*- 

a  =  2281"        -        X'  =  492"  N 


133  (See  Note,  p.  323) 


8h-    11^ 


a  =  1834 


a  =  1834"  ar.  co.  log.  6.73660 
X'  =    359  log.  2.55509 


6   =11"  4' 30"      tan.   9.29169 

Appar.  dist.  of  cen.      a  =  1869" 
Sum  of  semi-diam.     -    -   1846 


X'  =  359  N 

log.  3.26340 

ar.  CO.  cos.   0.00817 


-  loff.    3.27157 


487' 


23' 


22" 


t  =  l""-  2^ 


334  ASTRONOMY. 

8h.      11m. 

+  1 


2d  Approxi.    8'^-    12™- 

For  the  Approxiinate  Time  of  the  End. 

h  =  1556",   d  =  1958"  —  1839"  =  119" . 

1556"  :  119"  :  :  60"^-  :  ^  =  4™- .6. 

10'-  49™- 
—  5 


1st  Approxi. 

W^ 

44m. 

10:>- 

49m. 

-     a  = 

: 1925" 

- 

- 

X' 

=  357" 

S. 

Con 

rection 

44m. 

for  5'"- 
-     a  = 

132 

X' 

17 

=  340 

lOh- 

:1793 

S. 

a  = 

1793" 

Ar. 

CO.  log. 

6; 

r4642 

_ 

log.  3.25358 

X'  = 

340 

- 

-     log. 

2.53148 

tan.  9.27790    -  Ar.co.  cos.  0.00767 


Appar.  dist.  of  cen.  a  =  1825"     -        3.26125 
1839 

133"  :       14"  :  :  S-"-  :/  =  0'"-.5. 

IQh.   44m. 
0    .5 

2d  Approxi.     10'^-   44™- .5 

For  the  Approximate  Time  of  Greatest  Ohsrvration. 
Approx.  time  of  begin.      -       8*^    12"'- 
Approx.  time  of  end,        -      10     44 


2  )  18     56 


1st  Approxi.      -      9     28 


PROB.   XXX.    TO  CALCULATE  A  SOLAR  ECLIPSE. 


335 


For  the  True  Times  of  the  Phases. 

Approx.  time  of   Approx.  time  of  Approx.  time 

Bsgiiming.       Greatest  Obscur.  of  End, 

gh.  1.2m.             Qh.  28'"-  IQh-  44™- 

Sim's  longitude,  17-5°  26'    l'!o,      175°  29     6.8,  175°32' 12!6 

Do.  semi-diam.,           15  57.0,              15  57.0,  15  57.0 

Moon's  app.  long.  174  55  36.7,       175  27     7.7,  176     2  17.2 

Do.  app.  lat.                  5  45.3 N,            0  43.5S,  5  32.4S. 

Do.  auo'm.  semid.        14  48.0,              14  45.1,  14  41.7 


gh.  12^- 
9     28 
10    44 


1824".3 
119  .1 

1804  .6 


k 


1705".2 
1923  .7 


X' 


345".3Nl 

43  .5  si 

|332  .4  Si 


388".8 
288  .9 


1856".7 
1835  .0 


S 


1840".0 
1833  .7 


For  the  True  Time  of  Beginning. 


a 

1824' 

.3 

k     - 

1705 

.2 

n 

388 

.8 

6=- 

-8001 

.1 

X'  - 
-6  =  c  = 

345 

.3 

X'- 

=  8346 

.4 

SH- 

A   - 

3696 

.7 

S- 

-A   - 

—16 

.7 

n 

- 

T 

- 

76 

-  log.  3.26109 

-  log.  3.23178 
Ar.  CO.  I02:.  7.41028- 


log.  3.90315- 


Ar.  CO.  log.  6.07850 

-  log.  3.56781 

-  log.  1.22272- 
Ar.  CO.  log.  7.41028- 

-  log.  1.88081 
Const,  log:.  1.47712 


Corr.  of  approx.  time,  -|-43'-.4 

Approx.  time,         -     S'^-  12"™    0   .0 


log.  1.63724-1- 


True  time  of  begin.     8     12     43    .4,  in  Greenwich  time. 
Ditf.  of  merid.         -     4     56       4 


True  time  of  begin.     3     16     39   .4,  in  New  York  time. 


336 


ASTRONOMY. 

For  the 

True  Time  of  End. 

a     - 

-     18n4".6 

_         .         _         _ 

locr.  3.25638 

k     - 

-     1923  .7 

- 

loa.  3.28414 

n     - 

-      288  .9 
~  12016  .3 

Ar.  CO, 

.  log.  7.53925— 

b  = 

log.  4.07977— 

X'     - 

—  332  A 

X'+ 6-c=— 12348  .7        -        -  Ar.  CO.  log.  5.90838— 

S  -L  A     -    -     3668  .7        -        -        -        -     log.  3.56451 
S— A     -    -     —1  .3        -        -        -        -     log.  0.11394— 

n Ar.  CO.  log.  7.53925 — 

T        -        -      76m.    -        -        -        -  1.88081 

Const,  losf.  1.47712 


Corr.  of  approx.  time,  — ^^0  -     log.  0.48401- 

Approx.  time,       -      lO*--  44"'-    0  .0 


True  time  of  end,      10     43     57.0,    in  Greenwich  time. 
Diff.  ofmerid.       -       4     56       4 


True  time  of  end,        5     47     53,         in  New  York  time. 

For  the  True  Time  of  Greatest  Obscuration. 

True  time  of  beginning,      -        -        -     18^-  12*"  43«  .4 
Do.  of  end,       -        -        -        -     10     43     57    .0 


2)18     56     40    .4 
2d  Approx.     9     28    20    .2 


Diff. 


gh. 

49m. 

- 

- 

X'  = 

138"     S. 

9 

28 

x  = 

Diff 

43  .5S. 

F. 

21 

94  .5 

21™- 

:20^- 

:  :  94".5  :  1" 

.5 

28«- 

20''- 

43 

.5 

gh. 

X'  =  45 

.0 

PROB.    XXX.    TO    CALCULATE    A    SOLAR    ECLIPSE.  337 

1705".2      388".8 
1923  .7      288  .9 


k'  =  3628  .9  :  n'  =  G77  .7  : :  X'  =  45".0  :  a'  =  8".4 

Time  of  beginn.  8^-  12'»-  43^  .4,  at  9^-  28"^-  a  =  119".l 
Time  of  end,      10    43     57  . 0  a'  =     8  .4 


L  =    2     31     13  .6  m  =  110  .7 

3628".9  :  110".7  :  :  2'>-  31'^- 13^- .6  :  4^«  38'- .2 

9h-  28       0  .0 


True  time  (nearly)  9     32    38  .2 

21m. .  4m.  383.  .  .  94".5  .  20".9 
43  .5 


At9h-32'"-38^-,X'  =  64  .4 

3628".9  :  677".7  :  :  64".4  :  12".0  ;  at  9^-  32'"- 38^  a  =      8".4 

a'  =    12  .0 


m  =  — 3  .6 


3628".9  :  —  3".6  :  :  2^-  31™-  13^-. 6  :  —  9^  .0 

9^-  32™-  38  .2 


9     32     29  .2 


True  time  of  greatest  obscur.     -    9'^-  32'n-  29^-.2,inGreenw.time, 
Dilf.  ofmerid.  -        -        -    4    56       4 


True  time  of  greatest  obscur.     -    4    36     25  .2,  in  N.  Y.  time. 


For  the  Qxiantity  of  the  Eclipse, 

9^:  32™-  38'-     -    X'  =  64".4 
21™-  :^^- : :  94".5  :     0  .7 


At  nearest  approach  of  centres,     -    x'  =  65  .1 
«  «  «         -        -    a  =  12  ,0 

43 


338                                                    ASTRONOMY. 

a      -      12".0    -     Ar.  co.  log.  8.92082, 
X'      -      65  .1     -        -        -     1.81358 

-      log.  1.07918 

fl       -        -        -        -        tan.  0.73440, 

-     Ar.  CO.  cos.  0.74165 

Shortest  distance  of  centres,   -  66".2 

log.  1.82083 

Sum  of  semi-diameters,    -      1837.0 


1770  .8 
15'  57"  :  1770".8  :  :  6  :  11.1  digits  eclipsed. 

For  the  Situation  of  the  Point  at  lohich  the  Obscuration 
commences. 
S"^-  12m-    .      .a      =1824",       -        -        X' =  345".3  N. 
76™-  :  43^-  :  :  1705" :      16,  76>n- :  43^- : :  389" :      3  .7 


At  the  beginn.      -       a  =  1808, 

a 

-     1808    -    Ar.  CO.  log.  6.74275 

X' 

-      341.6         -        log.  2.53352 

&  = 

=  10°  41'  53"      -        tan.  9.27627 

X'  =  341  .6 


Obliq.  of  eclip.  (Prob.  X),  23°  27' 47"  sin.  9.60005  -  tan.  9.63753 
Sun's  longitude,       -       175    26    3    sin.  8.90093  -  cos.  9.99862 


sin.  8.50098,     tan.  9.63615 
Sun's  declination,  1°  49'  0";  Angle  of  pos.  23°  23'  59". 

Mean  time  of  begin.  3'^- 16™-  39^-,  Lat.  40°  42'  40",  Dec.  1°  49'  0" 
Equa.  oftime,        -  5     58  90  90 

Appar.time,        -       3   22     37,PZ  =  49    17  20,PS=88    11 
60 


4 )  202       37 


Hour  angle  P  =  50°  39'  15"   -    cos.  9.80210 
Co.  lat.  ^P  Z  =  49  17  20    -    tan.  0.06526 


m  =  36°  23'  0"   -    -    tan.  9.86736 
Co.  dec.  PS  =  88  11  0 


m'  =  51  48  0   -   Ar.  co.  sin.  0.10466 


PROB.  XXX.    TO  CALCULATE  A  SOLAR  ECLIPSE.  339 

m'=51  48  0  -  Ar.  CO.  sin.  0.10466 
w  =  36  23  0  -  -  sin.  9.77320 
P  =  50  39  15    -    .    tan.  0.08627 


S  =  42  38  10    -    -    tan.  9.96413 

Angle  of  position,  -        -        —23°  23' 50" 

Angle  from  eclip.  {&),       -        -         —  10  41  50 
Angle  of  dec.  circle  from  vertex  (S)j      42  38  10 

90 


Angular  dist.  of  point  first  touched  from  vertex,  98   32,  to  the  right. 

For  the  Beginning  and  End  of  the  Anmdar  Eclipse. 

Approx.  time,  ^^-  32'"-  29^.2  =  true  time  of  greatest  obscur. 
At  this  time,  a  =  12".2,  V  =  63".7. 

a  =  12".2     -      Ar.  co.  log.  8.91364      -        -        log.  1.08636 
X'=63.7    -        -  W.  1.80414 


&  =  79°  9'  30"      -  tan.  0.71778     -     Ar.  co.  cos.  0.72564 


A  =  64".9        -        -        log.  1.81200 

S  +  A  =  135".8  -  log.  2.13290,   &  =  79°  9'  30"  -  Ar.  co.  sin.  0.00783 
S  —  A  =     6  .2  -  log.  0.79239,  k=  3628.9         -        loa.  3.55977 


2)2.92529,   h     -        -       Ar.  co.  log.  6.43240 

1.46264  ...        -        1.46264 

L  =  152™-  -        -        -  log.  2.18184 

Const,  log.  1.77815 

;  =  0'^-  1"^  11^6         -  log.  1.85503 

Time  of  greatest  obscur.     -    4  36    25  .2 

Formation  of  ring,     -       -    4  35    13  .6,  New- York  time. 

Rupture  of       do.      -       -    4  37    36.8  «        « 


APPENDIX. 


TRIGONOMETRICAL    FORMUL./E. 


I.  Relative  to   a   Single  Arc  or  Angle   a. 

1.  sin2  a  +  cos2  a  =  1 

2.  sin  a  =  tan  a  cos  a 

Q        o,v,  tan  a 

o.       sin  a 


10. 


v/1  +  tan^  a 


4,       cos  a  = 


\/  1  +  tan^a 
sin  a 
cos  a 

1  cos  a 


5.  tan  a  = 

6.  cot  a  — 

tan  a      sin  a 

7.  sin  a  =  2  sin  ^  a  cos  ^  a 

8.  cos  a  =  1  —  2  sin^  i « 

9.  cos  a  =  2  cos^  ^  a  —  1 

+„v,  1                 sin  a 
tan  ^   a  — ■  

1  +  cos  a 


11.       cot  ^  a 


sin  a 


1  —  cos  a 


12.  tan^  i  a  =  ^  "  ^^^  ^ 

1  +  cos  a 

13.  sin  2  a  =  2  sin  a  cos  a 

14.  cos  2  a  =  2  cos2  a  —  1  =  1  —  2  sin^  a 


342  APPENDIX. 

II.  Relative  to  Two  Arcs  a  and  b,  of  which  a  is  supposed  to 
he  tlie  greater. 

15.  sin  (a  +  6)  =  sin  a  cos  6  +  sin  6  cos  a 

16.  sin  {a  —  h)  -  sin  a  cos  b  —  sin  6  cos  a 

17.  cos  (a  +  6)  =  cos  a  cos  b  —  sin  a  sin  6 

18.  cos  (a  —  6)  =  cos  a  cos  h  +  sin  a  sin  6 

in        *      /      I    i,\  tan  a  +  tan  6 

19.  tan  (a  +  o)  =  ' 

1  —  tan  a  tan  6 

on        *       /          z.\        tan  a  —  tan  b 

20.  tan  (a  —  h)  - - 

1  +  tan  a  tan  b 

21.  sin  a  +  sin  6  =  2  sin  ^  («  +  6)  cos  \  [a  —  b) 

22.  sin  a  —  sin  6  =  2  sin  \{a  —  h)  cos  \  {a  -f  6) 

23.  cos  a  +  cos  b  -2  cos  ^  («  -j-  b)  cos  |^  («  —  6) 

24.  cos  6  —  cos  a  =  2  sin  ^  (a  +  h)  sin  ^  (a  —  6) 

or        .  1*1.       sin  (a +  6) 

25.  tan  a  +  tan  6  =  L_  J_>L 

cos  a  cos  b 

oc        .  ♦       z,        sin  (a  —  b) 

26.  tan  a  —  tan  b  =  !^ L 

cos  a  cos  0 

27.  cota+cot6=?iLL«4Li) 

sni  a  SHI  6) 

oo  *  I,  *  sin  (rt  —  b) 

28.  cot  6  —  cot  a  =  -^-^^ — . ' 

Sin  a  Sin  6 

sin  «  +  sin  6  _  tan  \{a-\-h) 
sin  a  —  sin  6      tan  h  (a  —  6) 

cos  b  +  cos  a  _  cot  ^  {a  +  b) 
cos  b  —  cos  a      tan  ^  (a  —  6) 

tan  a  +  tan  6  _  cot  b  +  cot  a  _  sin  (a  +  b) 
tan  a  —  tan  b      cot  6  —  cot  a      sin  (a  —  6) 

cot  b  —  tan  a  _    cot  a  —  tan  b    _  cos  {a  +  6) 
cot  b  +  tan  a       cot  a  +  tan  6        cos  (a  —  i) 

33.  sin=  a  —  sin^  b  -  sin  (a +6)  sin  (a —  6) 

34.  cos-  a  —  sin 2  b  —  cos  {a  +  6)  cos  (a  —  6) 

35.  1  ±  sin  a  =  2  sin^  (45°  ±  \  a) 

36.  i^^-Hl_^=  tan^  (45°  ±  i  a) 
1  T  sin  « 


29. 
30. 
31. 
32. 


37. 
38. 
39. 
40. 

41. 

42. 
43. 
44. 
45. 


TRIGONOMETRICAL    FORMULAE. 

1  ±  sin  a 


343 


tan  (45°  ±  l  a) 
cos  a 

1  —  sin  a  _  sin2  (45°  —  \  a) 
1  —  cos  a  sin  ^  \  a 

1  +  sin  6  ^  sin=^  (45°  ^  \h) 
1  +  cos  a 

1  +  tan  h 


tan  h 
tan  & 


cos^  ^  a 
-  tan  (45°  +  h) 

-tan  (45°  — 6) 


46. 


1  +  tan  6 

sin  a  cos  6  =  ^  sin  (a  +  6)  -|-  -^  sin  (a  —  6) 
cos  a  sin  h  =\  sin  (a  +  ^)  —  \  sin  (a  —  6) 
sin  a  sin  6  =  ^  cos  [a  —  b)  —  i  cos  (a  -f-  6) 
cos  a  cos  b  =  ^  cos  (a  +  6)  +  ^  cos  (a  —  b) 

III.   Trigonometrical  Series, 
a^     ,         a^' 


sin  a  =  a  — 


+ 


cos  a 


+ 


2.3 

tan  a  =  a-{-  -^—  + 


2.  3.  4.  5 

2.3.4 

3.5 


<fec. 


2.  3.  4. 5.  6 
17a 


+  &C. 


,  \       a  a^ 

cot  a  =   -  —  -  — 

a       3         3-.  5 


3^5.7 
2a' 


4-(fec. 


3^5.7 


—  —  (fee. 


Let  a  =  length  of  an  arc  of  a  circle  of  which  the  radius  is  1, 
and  (a")  =  number  of  seconds  in  this  arc,  then  to  replace  an  arc 
expressed  by  its  length,  by  the  number  of  seconds  contained  in  it, 
we  have  the  formula, 

47.  a  =  {a")  sin  1";  log.  sin  1"  ="6.685574867. 

IV.  Differences  of  Trigonometrical  Lines. 

48.  A  sin  a;  —  -{-  2  sin  ^  A  x.  cos  {x  -\-\i\  x) 

49.  A  cos  X  -  —  2  sin  ^  A  x.  sin  (.r  +  ^  a  x) 

sin  A  X 


50.      A  tan  x=  ■\- 


cos  X.  cos  [x  +  A  ;r) 


344 


APPENDIX. 


51. 


A  cot  X  =  — 


Sin  A  X 


sin  X.  sin  {x  +  a  x) 


V.  Resolution  of  Right  Angled  Spherical  Triangles. 
Given.  Required.  Solution. 

Hypothen.  [  ^^^®  ^P-  S^^-  '^"°-  ^^      ^^"  ^'  "  ^^"^  ^^ '  ^"^  " 
and       {  side  adj.  civ.  ang.  53      tan  a*  =  tan  h  .  cos  a 

an  angle    j  ^j-^g  other  an^le      54      cot  x  =  cos  h  .  tan  a 


the  other  side         55      cos  x  = 


cos  h 


cos  5 


Hypothen.  I 

and       {  ang.  adj.  giv.  side  56      cos  x  —  tan  5 .  cot  h 

a  side      I 

ang.  op.  giv.  side  57      sin  x 


\  the  hypothen. 
A  side  and  j 

the  angle  {  the  other  side 
opposite 


sni  5 
sin  A 

sin  s 


58  sin  X 

sni  a         I  S 

1  § 

59  sni  X  —  tan  s .  cot  a  y  go 


the  other  ano;le      60      sin  .v  = 


cos  a 


Asideandfth^^^yPothen. 

the  angle  <J  the  other  side 
adjacent    j^  ^^^^  ^jj^g^.  ^^j-^g^g 

f  the  hypothen. 
The  two  ! 

sides       I  , 

Lan  angle 

the  hypothen. 


cos  s         J   ^ 

61  cot  X  =  cos  a  .  cot  5 

62  tan  x  =  tan  a  .  sin  s 

63  cos  X  =  sin  a  .  cos  s 

64  cos  X  =  rectan^.  cos.  of  the 


The  two 

angles 


side 


65 
66 

67 


giv.  sides 

cot  X  —  sin  adj.  side  x  cot. 

op.  side 

cos  X  =  rectang.  cot.  of  the 

given  angles 


cos  2' 


,  _  COS.  opp.  ang. 


sin.  adj.  ang. 


In  these  formulse,  x  denotes  the  quantity  sought. 
a  —  the  given  angle 
5  =  the  given  side 
h  =  the  hypothenuse. 


RESOLUTION    OF    SPHERICAL    TRIANGLES.  345 

The  formulae  for  the  resolution  of  right  angled  spherical  trian- 
gles are  all  embraced  in  two  rules  discovered  by  Lord  Napier,  and 
called  Napier's  Rules  for  the  CircAilar  Parts.  The  circular 
parts,  so  called,  are  the  two  legs  of  the  triangle,  the  complement 
of  the  hypothenuse  and  the  complements  of  the  acute  angles. 
The  right  angle  is  omitted.  In  resolving  a  right  angled  spherical 
triangle,  there  are  always  three  of  the  circular  parts  under  consid- 
eration, namely,  the  two  given  parts  and  the  required  part. 
When  the  three  parts  in  question  are  contiguous  to  each  other, 
the  middle  one  is  called  the  middle  part,  and  the  others  the  adja- 
cent jiarts.  When  two  of  them  are  contiguous,  and  the  third  is 
separated  from  these  by  a  part  on  each  side,  the  part  thus  sepa- 
rated is  called  the  middle  part,  and  the  other  two  the  opposite 
parts.  The  rules  for  the  use  of  the  circular  parts  are  (the  radius 
being  taken  =  1), 

1.  Sine  of  the  middle  part  =  the  rectangle  of  the  tangents  of 
the  adjacent  parts. 

2.  Sine  of  the  middle  part  ==  the  rectangle  of  the  cosines  of  the 
opposite  parts. 

Equations  52  to  67,  are  sufficient  to  resolve  all  the  cases  of 
right  angled  spherical  triangles  ;  but  they  lack  precision  if  the 
unknown  quantity  is  very  small  and  determined  by  means  of  its 
cosine  or  cotangent ;  or,  if  the  unknown  quantity  is  near  90°, 
and  given  by  a  sine  or  a  tangent :  in  these  cases  the  following  for- 
mulae may  be  used, 

68.  tan=ia  =  -^(l±C) 

cos  (B  —  C) 

69.  tan^  ^^^sm[a-c)^ 

sin  (a  -\-  c) 

70.  tan^  ^  c  =  tan  \  [a -[- h)  ton  \  {a  —  b), 


i|,      71.    tan  (45°  —  ^b)=  %/  tan  (45°  —  x),  tan  a:  =  sin  a  sin  B. 
72.     tan^  ^  5  =  tan  (^  ""  ^  -f  45°)  tan  (?-^  —  45°) 

a  is  the  hypothenuse,  B,  C,  the  acute  angles,  and  b,  c,  the 
sides  opposite  the  acute  angles. 

44 


346 


APPENDIX. 


VI.  Resolution  of  Oblique  Angled  Spherical  Triangles. 
If  A,  B,  C,  denote  the  three  angles  of  a  spherical  triangle,  and  a 
6j  c,  the  sides  which  arc  opposite  to  them  respectively, 
sin  A        sin  B      sin  C 
sin  a 


73. 


b  sin  c 

or,  the  sines  of  the  angles  are  proportional  to  the  sines  of  the 
opposite  sides. 

74.  cos  c  =  cos  a  cos  6  +  sin  a  sin  b  cos  C 

75.  cos  c  =  cos  (a  —  b)  —  2  sin  a  sin  b  sin^^  C 

76.  cos  C  =  sin  A  sin  B  cos  c  —  cos  A  cos  B 

77.  sin  a  cos  c  =  sin  c  cos  a  cos  B  +  sin  ,6  cos  C 

78.  sin  a  cot  c  =  cos  a  cos  B  +  sin  B  cot  C 

79.  sin  a  cos  B  =  sin  c  cos  b  —  sin  b  cos  c  cos  A 

I.   Given  the  three  sides,  a,  b,  c. 
To  find  one  of  the  angles. 


80. 

81. 

82. 


sni 


o  1  A  _  sin  [k  —  b)  sin  {k  —  c) 


sin  b  sin  c 

^  1   .     sin  A:  sin  (k  —  a) 

cos  2  i  A  = ^ 

sin  b  sin  c 

2k=a  +  b  +  c 


II.   Given  the  three  angles  A,  B,  C. 
To  find  one  of  the  sides. 

cos  K  cos  (K  — 'A) 


83 


sin 


H« 


sin  B  sin  C 


84.      cosHa=^'(^^-^^'"^^^^^^ 


85.      2  K  =  A  +  B 


sin  B  sin  C 
C. 


iW-  Given  tioo  sides  a  and  b,  and  the  included  angle  C. 


1°.  To  find  the  two  other  angles  A  and  B. 


86. 


87. 


tan  A  (A  +  B)  =  cot  A  C.  S^UlL"^. 

cos  i  fa 


^) 


{a-\-b) 

tan  HA  -  B)  =  cot  ^  C.  !!^-ii^— t-) 
^  ^  ^  ^      sm  ^  (a  +  b) 


Napier's 
'  Anuiosries 


RESOLUTION  OF   SPHERICAL  TRIANGLES.  347 

2".  To  find  the  third  side  c. 


1  *       1  /  T\  sin  i  (A  +  B) 

tanic  =  tanH«-^).,j^|^^ 


tan  i  c  =  tanJ^  (a  +  b).^^liJA±3 
^  ^^    ^    ^  cos^(A  — B) 

or  equa,  74, 

IV.   Given  two  angles  A  ajid  B,  and  the  adjacent  side  c. 
1°.  To  find  the  other  two  sides,  a  and  b. 

80.    tan    1  (a  +  Z»)  =  tan  h  c.  c_£!i_(ArZ^)  ] 

"       cos  |(A  +  B)  I        Napier's 

90.    tan  h  (a-b)=  tan  l  c.  siii_i^^^)  |      Analogies, 

-   ^  ^  ^      sini(A  +  B)J 

2°.  To  find  the  third  ande  C. 


91. 


cot  i  C  =  tan  i  (A  -  B).  ^."\^ /^±^) 
^  ^  ^  sin  i  (a  —  6) 

cot^C  =  tani(A+B).^-°4(;+A) 

cos  ^(a  — 6) 


or  equa.  76. 

V.   Given  two  sides  a,  b,  and  an  opposite  angle  A. 
To  find  the  other  opposite  angle  B  ;  take  equation  73,  or  the 
proportion  ;  sines  of  the  angles  are  as  sines  of  the  opposite  sides, 
(For  the  methods  of  determining  the  remaining  angle  and  side,  see 
page  348,  Case  3.)  •   ' 

VI,   Given  two  angles  A,  B,  and  an  opposite  side  a. 
To  find  the  other  opposite  side  b  ;  sines  of  the  angles  are  pro- 
portional to  the  sines  of  the  opposite  sides.     (For  the  methods  of 
determining  the  remaining  side  and  angle,  see  page  349,  Case  4), 

Other  Methods  of  Resolving  Oblique  Angled  Spherical 
Triangles. 

Except  when  three  sides  or  three  angles  are  given,  the  data 
always  include  an  angle  A,  and  the  adjacent  side  6,  besides  a 
third  part.  The  required  parts  in  the  different  cases  may  be 
found  by  the  following  formulae,  and  formula  73. 

92.    tan  m  =  tan  6  cos  A  93.    cot  n  =  tan  A  cos  b 


348 


:8 

APPENDIX. 

94. 

c    = 

m  +  m' 

95. 

C  = 

n  -\-  n' 

96. 

cos  a 

cos  m' 

97. 

cos  A  _ 
cos  B 

sin  n 

cos  b 

cos  m 

sin  n' 

98. 

tan  A  _ 
tan  B 

sin  m' 

99. 

tan  a  _ 

cos  w 

sin  wi 

tan  h 

cos  w' 

100. 

sin 

k 

=  sin  A 

sin  b. 

From  the  angle  C,  (Fig.  79),  a  perpendicular  C  D  is  let  fall 
upon  the  opposite  side  c,  which  divides  the  triangle  into  two  right 
angled  triangles,  that  are  resolved  separately.  In  the  one,  A  C  D, 
A  and  b  are  known,  and  it  is  easy  to  find  the  other  parts,  which, 
joined  to  the  third  given  part,  serve  to  resolve  the  second  right 
angled  triangle  BCD,  and  determine  the  unknown  quantity 
required,  m,  m'  denote  the  two  segments  of  the  base ;  w,  n'  the 
two  parts  of  the  angle  C ;  and  k  the  perpendicular  arc  C  D. 

It  must  be  observed,  that  if  the  perpendicular  C  D  fell  without 
the  triangle,  m  and  m',  n  and  n'  would  have  contrary  signs :  this 
happens  when  the  angles  A  and  B  at  the  base  are  of  different 
kinds,  (the  one  Z.,  the  other  >90°).  When  it  is  not  known 
whether  this  circumstance  has  place  or  not,  the  problem  is  sus- 
ceptible of  two  solutions. 

The  perpendicular  arc  is  to  be  let  fall  from  that  one  of  the  ver- 
tices B  or  C,  for  which  it  does  not  divide  into  two  parts  the  3d 
element  given  with  A  mid  b. 

The  detail  of  the  different  cases  is  as  follows :  the  data  are  A, 
6,  and  another  arc  or  angle. 

Case  1.   Given  two  sides  and  the  included  angle,  b,  c,  A. 

Equation  92  makes  known  m,  94  m',  which  may  be  negative, 
(what  the  calculation  shows),  96  a,  98  B,  and  equation  73,  (page 
346),  C,  which  is  known  in  kind. 

Case  2.   Given  two  angles  and  the  adjacent  side,  A,  c,  b. 

Equation  93  makes  known  w,  95  n\  which  may  be  negative, 
(what  the  calculation  shows),  97  B,  99  a ;  finally,  equation  73 
(page  346),  gives  C,  which  is  known  in  kind. 

Case  3.   Given  two  sides  and  an  opposite  angle,  b,  a,  A. 

Equation  92  gives  w,  96  771',  94  c,  98  and  73  B  and  C ; 

or  else,  93  gives  w,  99  n',  95  C,  97  and  73  B  and  c. 

This  problem  admits  in  general  of  two  solutions.    In  effect,  the 


PARALLAX  IN  RIGHT  ASCENSION  AND  DECLINATION.         349 

arc  m'  or  v!  being  given  by  its  cos.,  may  have  either  the  sign 
+  or  — ;  there  are  then  two  vahies  for  c,  and  also  for  C.  m'  and 
n!  enter  into  equations  97  and  98  by  their  sines,  whence  result  two 
values  of  B ;  same  for  C  and  c. 

Case  4.   Given  two  angles  and  an  opposite  side,  A,  B,  b. 
Equation  92  gives  m,  98  m',  94  c,  96  a,  and  equation  73  makes 
known  C ; 
or  else  93  gives  n,  97  n',  95  C,  99  and  73  a  and  c. 

There  are  also  two  solutions  in  this  case  ;  for,  m'  or  n'  is  given 
by  a  sin.,  and  therefore  two  supplementary  arcs  satisfy  the  ques- 
tion. Thus  c  in  94,  or  a  in  99,  receives  two  values ;  same  for  a 
in  96  and  c  in  95,  &.c. 

When  the  triangle  is  isoceles,  B  =  C,  6  =  c,  the  perpendicular 
arc  must  be  let  fall  from  the  vertex  A,  and  the  equations  become 
very  simple.     We  find 

101.  sin  1  a  =  sin'i  A  sin  b 

102.  tan  I  a  =  tan  b  cos  B 

103.  cos  b     =  cot  B  cot  i  A 

104.  cot  i  A  =  cos  1  a  sin  B 

The  knowledge  of  two  of  the  four,  elements  A,  B,  a,  b,  which 
form  the  isoceles  triangle,  is  sufficient  for  the  determination  of 
the  two  others. 


INVESTIGATION  OF  ASTRONOMICAL  FORMULAE. 

FornmlcB  for  the  Parallax  in  Right  Ascension  and  Declina- 
tion^ and  in  Longitude  and  Latitude.  (Referred  to  from 
Article  106,  page  47). 

Let  5  (Fig.  80)  be  the  true  place  of  a  star  seen  from  the  centre 
of  the  earth,  s'  the  apparent  place,  seen  from  a  point  on  the  sur- 
face of  which  z  is  the  zenith,  the  latitude  being  /.  The  displace- 
ment s  s'  =  p  is  the  parallax  in  altitude,  which  takes  effect  in  the 
vertical  circle  z  s'  ]  p  is  the  pole  ;  the  hour  angle  z  p  s  -  qis 
changed  into  z  p  s',  and  s  p  s'  —  a.  is  the  variation  of  the  hour 
angle,  or  the  parallax  in  right  ascension ;  the  polar  distance 


350 


APPENDIX. 


p  s  =  d  is  changed  into  p  s' ;  the  difference  5  of  these  arcs  is  the 
parallax  in  declination  or  of  polar  distance.  We  have  (For.  73, 
p.  346), 

sin  s'  :  smp  s  (d)  : :  sin  sp  s'  (a)  :  sin  ss'  (/>), 

sins  p  s'  {q  +  a):  sin  z  s'  (Z) : :  sin  s'  :  sinpz  (90°  —  I). 

Multiplying,  term  by  term,  we  obtain, 

sin  s'  sin  (^  4-  «) :  sin  d  sin  Z  :  :  sin  a  sin  s'  :  sin  p  cos  I ; 

V  •  sin  p  cos  ^    .    ,     ,     ^ 

whence,        sm  a  =  — L_ sm  (o  +  a) 

sin  d  sin  Z 

Or,  substituting  for  p  its  value  given  by  equ.  (10,)  p.  43,  and  re- 
placing H  by  P, 

„• ,  sin  P  cos  I   ■    ,     ,      s  f  A\ 

sm   a  = . sm  (q  4-  a).  .  .  .  (A). 

smd  ^'i^    J  ^    ^ 

This  equation  makes  known  a  when  the  apparent  hour  angle 

z  p  s'  =  q  -\-  a,  seen  from  the  earth's  surface,  is  given  ;  but  if  we 

know  the  true  hour  angle  z  p  s  —q,  seen  from  the  centre  of  the 

earth,  developing  sin  {q  +  a),    (For.  15,  p.  342),  and  putting 

sin  P  cos  I 

: =  m 

sm  d 

sin  a  =  m  (sin  q  cos  a  +  sin  a  cos  q), 

or,  dividing  by  sin  a, 

1  =  m  (sin  ^  cot  a  +  cos  q)] 

whence,  by  transformation, 

w,  sin  a  .  . 

tan  a  = 1 —  =  m  sm  o  +  m^  sm  q  cos  q. 

1  —  *?t  cos  q 

Restoring  the  value  of  w, 

sin  P  cos  I   .  ,    /sin  P  cos  l\ ,    . 

tan  a  = sm  <7  +  I I  -  sm  q  cos  q. 

sin  d  >     sin  c/      / 

Patting  the  arc  a  in  place  of  its  tangent,  and  P  in  place  of  sin  P, 
and  expressing  these  arcs  in  seconds,  (For.  47,  p.343),  there  results, 

P  cos  Z     •  ,       /PCOSZV,      •  .       -,,,  ,r)\ 

a  =   sin  r/  +   8    -. — —I-  sm  q  cos  q  sm  1"  .  .  (B). 

sin  c/  ^        \  sm  d  f  ^        ^  ^    ^ 

The  parallax  in  declination  S  is  the  difference  of  the  arcs  p  s 
=  d,  ps'  =  d  +  6.  Let  zs  =  z,  and zs'  =  Z.  The  triangles  z p  s 
and  zps'  give  (For.  74  and  73), 


PARALLAX    IN    RIGHT    ASCENSION    AND    DECLINATION.    351 

-„  COS  f/  —  sin  Z  COS  2;     cos  (d  4- 6)  —  sin  Z  cos  Z 

1°.       cos  p   Z  S   —    r— =   ^^ —-' ; , 

COS  I  sin  z  cos  Z  sin  Z 

on      •  sin  d  sin  a     sin  (d  +  ^\  sin  (q  4-  a) 

2°.    sm  p  z  5  =  i  = i — ' — il AU — i . 

sin  z  sin  Z 


From  the  first  equation  we  derive, 
N^  _  COS  d  sin  Z  —  sin 
sin; 
_  COS  d  sin  Z —  sin  I  (cos  z  sin  Z  —  sin  z  cos  Z) 


/,  ,    rN,      cose?  sin  Z  —  sin  Z  cos  2;  sin  Z    ,     ■     ,         r, 
cos  [d-\-  0)  = 4-  sin  Z  cos  Z 

sin  z 


sin  2; 
cos  c?  sin  Z  —  sin  Z  sin  (Z  —  2;) 


sin  z 
or,  (equ.  10,  p.  43), 

sin  Z 


(cos  d  —  sin  P  sin  Z) ; 


sm  2; 

from  the  second, 

sin  Z      sin  {d  +  ^)     sin  (^  +  a)  . 

sin  z  ~      siad  sin  ^ 

substituting, 

.  ,   ,    ..      sin  (c?  +  5)     sin(<7  4-aV        ,        •    -o    ■    tn 

cos  (cZ  +  ^)  =  — ^ ^  .  i-J —  (cos  d  —  sin  P  sin  I) 

sin  d  sin  q 

cos  (cZ  +  6)  _sin  {q  +  «)    /cos  d  sin  P  sin  l\ 

sin  (cZ  +  ^)  sill  q        *  sin  cZ  sin  d     / 

^  ,  ,  ,    rx      sin  («  +  a)  /     .     ,       sin  P  sin  Zv    //-in 

cot  ( cZ  +  (5)  = \^  ^    ^  I  cot  d—  : — ; —  I.  (C) 

sm  q        ^  sin  a      / 

_  ^  ^  sin  P  sin  Z 

Put  tan  X  — . — ; —  ; 

sin  d 

then,   cot  {d-{-h)  =  ^'^  [  ^  +— ^  (cot  d  —  tan  x) 
sm  q 

_  sin  {q  -}-  a)  /cos  d sin  x\ 

sin  q        *  sin  d       cos  :r' 

_  sin  {q  +  a)     cos  cZ  cos  x  —  sin  cZ  sin  x 
sin  q  sin  cZ  cos  x 

_  sin  (^  -J- «)  cos  (cZ  +  x)  /j^s 

sin  ^f  sin  fZ  cos.  x 


352  APPENDIX. 

The  apparent  polar  distance  (d  +  (5)  being  computed  by  either 
of  the  formulae  (C)  and  (D),  we  have  o  -  (d -}■  o)  —  8. 

Formulae  may  be  obtained  that  will  give  the  parallax  in  latitude 
without  first  finding  the  apparent  latitude  (except  approximately.) 

From  equa.  (C)  we  obtain, 

sin  P  sin  Z  _  ,        sin  7  cot  {d.  4-  5) 

sm  d  sm  {q  -\-  a) 

and  we  also  have, 

„.    y  ,  /  ,  ,    J,       cos  d      cos  (d  4-  0)  sin  5 

cot    d  —  cot   [d  +   6)   = ^^ -L-/  := : 


sin  d      sin  {d  +  0)      sin  rf  sin  (tZ  +  5)  ' 
the  sum  of  these  equations  gives 

sin  P  sin  Z^^^^       +  ^)  (1  -  __!i!Li_\  + ^ 

sm  d  \  sin  [q -\-  o.)f      sin  d  sin  {d  +  0) ' 

Now,  1  —     _!!£-l__-  =  s"^  (?  +  «)—  sin  q 

sin  [q  +  a)  sin  {  q  -[-  a) 

^  2  sin  ^acos  {q+\(t)  ^  sin  a  cos  (y  +  j  «)     .p^j.  22  13  ) 
sin(5'+a)  sin  (g*  4- a)  cos  ^  a 

cos  (</  +  i  a)  sin  P  cos  I     ,  , .  ^ 

= ^-^  .'    ^ — ^ ,  by  equa.  (A). 

sm  d  cos  ^  a 

Substituting, 

sin  P  sin  Z  wj   ,    ^n  cos  (<7  -[-  i  a)  sin  Pcos  Z  , 

: =  cot  [d  +  d)  ll_ \-^-^ •  + 

sm  d  sm  d  cos  ^  a 

sin  ^ 
sin  <Z  sin  (rf  +  6)' 
or,  sin  5  =  sin  P  sin  Z  sin  (<£  +  5)  — 

cos  (d  +  (5)  cos  (5^  +  i  «)  sin  P  cos  Z      .-pv 
cos  ^  a 
=  sin  P  sin  I  [sin  (cZ  +  (5)  —  tan.  1/  cos  ((Z  -\-  »5)], 

,  .  .  cot  Z  cos  (7  +  i  a) 

makmg  tan  2/  =  \^-^-^ —  ■> 

cos  ^  a 

whence,         sin  S  =  ^'"  ^  ^^^  ^  sin  (cZ  +  <5  —  y)  .  .  .  .  (F) 
cos  y 

To  facilitate  the  calculation,  the  sines  of  6  and  P  in   eqs.  (E) 
and  (F),  may  be  replaced  by  the  arcs. 


LONGITUDE  AND  ALTITUDE  OP  THE  NONAGESIMAL.  353 

To  obtain  an  expression  for  the  parallax  in  declination  in  terms 
of  the  true  declination,  develope  sin  {d  -\-  6  —  y)  in  equation  (F), 
which  gives, 

.     ,      sin  P  sin  I  r  •    r  i   ,    ^\ 

sin  6  =  [sm  (a  +  (5)  cos  w  —  sin  y  cos  (d  +  o\\  • 

cos  y  I        ^  ^         V      I     yj ) 

developing  sin  {d  +  5)  and  cos  {d  +  h)  and  reducing,  we  have, 

.     .      sin  P  sin  Z  r  •     /  ,  s  .   ,  ,  ,         x    •     ^ 

sin  d  = [sin  [d  —  y)  cos  h  +  cos  {d  —  y)  sin  h\, 

cos  y  ^  /  J) 

dividing  by  cos  (5 , 

.        r        sin  P  sin  /  r  ■    /  7         \    ,  /  , 

tan  (5  = [sin  {d  —  y)-\-  cos  {d  —  y)  tan  h\ 

sin  P  sin  I 

■ .  sin  id  —  ?/) 

cosy  ^         ^' 

whence  tan  h  =  - 


sin  P  sin  I  ^^^  ,  j         . 
_  cos  (f/  —  y) 


cos  y 


■     /  7         N   ,    /sin  P  sin  l\ 

sm  {d  —  y)+  \ 1 

^     cos  u      f 


cos  y  ^     cos  y 

sin  [d  —  y)  cos  [d  —  y) ; 

or,  replacing  tan  ^  and  sin  P,  by  (5  and  P  ;  expressing  these  arcs 
in  seconds,  (For.  47.  p.  343),  and  reducing  by  For.  13,  p.  341, 

,       P  sin  Z    .     ,  ,  ,     ,    /P  sin  Z\2  sii-j   \»     .  \  ,r^\ 

(5  = sin  id  —  v)  +1  I  — =^ sin2(fZ — ^y).(G) 

cos  y  ^  ■y^    '    \  cosy  /       2  \         Ji  \    I 

If  the  place  of  a  body  be  referred  to  the  ecliptic,  similar  formu- 
las will  give  the  par aJla.v  in  longitude  and  latitude,  hut  as  the 
ecliptic  and  its  pole  are  continually  in  motion  by  virtue  of  the  di- 
urnal rotation  of  the  heavens,  it  is  necessary,  in  order  to  be  able 
to  determine  the  parallax  in  longitude  at  any  given  instant,  to 
know  the  situation  of  the  ecliptic  at  the  same  instant. 

This  is  ascertained  by  finding  the  situation  of  the  point  of  the 
ecliptic  90°  distant  from  the  points  in  which  it  cuts  the  horizon, 
and  which  are  respectively  just  rising  and  setting,  called  the 
Notiagesimal  Degree,  or  the  Nonagesimal. 

Let  K  (Fig.  81)  be  the  pole  of  the  ecliptic/6,  p  the  pole  of  the 

equator/ a;  /is  the  vernal  equinox,  the  origin  of  longitudes  and 

of  right  ascensions ;  h  b  s  is  the  eastern  horizon,  b  the  horoscope, 

or  the  point  of  the  ecliptic  which  is  just  rising ;  pz  =  90°  —  /  (the 

45 


354  APPENDIX. 

latitude  of  given  place) ;  K  p  =  u  the  obliquity  of  the  ecliptic. 
The  circle  K  z  n  v  is  at  the  same  time  perpendicular  at  n  to  the 
ecliptic  /  b,  and  at  v  to  the  horizon  h  b :  it  is  a  circle  of  latitude 
and  a  vertical  circle,  since  it  passes  through  the  pole  K  and  the 
zenith  z  ;  6  is  90°  from  all  the  points  of  the  circle  K  7i  v  ;  z  n  is 
the  latitude  of  the  zenith,  /  n  its  longitude ;  the  point  n  is  the 
nonagesimal,  since  6  n  =  90°  ;  n  v  is  the  altitude  of  this  point,  and 
the  complement  o(  z  n;  n  v  measures  the  inclination  of  the 
ecliptic  to  the  horizon  at  the  given  instant,  or  the  angle  b,  so  that 
b  =n  V  —  K  z ;  thus  fn  —  1^  the  longitude  of  the  nonagesimal, 
and  n  V  =  h  the  altitude  of  the  nonagesimal.  designate  the  situa- 
tion of  this  point,  and  consequently  ascertain  the  position  of  the 
ecliptic  and  its  pole  at  the  moment  of  observation. 

The  points  m  and  d  are  those  of  the  equator  and  ecliptic  which 
are  on  the  meridian ;  the  arc  /  m,  in  time,  is  the  sidereal  time  5, 
which  is  known  ;  the  arc/  i  =  90°,  since  the  plane  K  p  i,  passing 
through  the  poles  K  and  />,  is  at  the  same  time  perpendicular  to 
the  ecliptic  and  to  the  equator ;  the  arcm  i  =fi  — /  m  =  90°  —  s  ; 
then  the  angle  zpK--  180°  —  zpi  =  180°  —  mi=  90°  +  s. 

Now,  in  the  spherical  triangle  p  K  2;  we  know  the  sides  K 
p  =  u,  z  p  =  90°  —  l  —  H,  and  the  included  angle  z  pK  =  90°  -f  s ; 
and  may  therefore  find  K  z  =  hthe  altitude  of  the  nonagesimal, 
and  the  angle  pK  z  =  n  c^fc  — fn  =  90°  —  N,  complement  of 
the  longitude  N  of  the  nonagesimal.  Let  S  =  sum  of  the  angles 
K  z  p  and  zKp,  then  (For.  86,  page  346), 

tan  ^  S  =  —  tan  (180°  —  ^  S),  tan  ^  (90°  —  s)  =  _  tan 

(5  —  90°) ; 

substituting,  and  denoting  (180°  —  ^  S)  by  E,  we  have, 

tan  E  =  cos  h  (^IZ^  tan  i  (s  —  90°) .  .  .  (H). 
cos  -i-  (H  +  w)         ^  ^  ^        ^    ^ 

Again,  let  D  =  Z  K  />  —  K  z  p,  then  (For.  87), 

tan  i  D  =  ?|!li-(l^)  cot  ^  (90°  +  5); 
sm  ^  (H  +  w) 


PARALLAX    IN    LONGITUDE    AND    LATITUDE.  355 

whence,  by  transforming  as  above,  and  denoting  (180°  —  ^  D) 
by  F,  we  have, 

Sin  ^  [t\  -\-  u) 
Now,  |S+|D=jt>Kz  =  90°  —  N; 

whence,  N  =  90°  —  (i  S  +  ^  D), 

or, 

N  =  360°4-90°  — (^S+^D)  =  180°  — ^S  +  180°  — ^D  +  90°, 
consequently,  N  =  E  +  F  +  90°  .  .  .  (J), 

rejecting  360°,  when  the  sum  exceeds  that  number. 

Next,  for  the  altitude  of  the  nonagesimal  we  have,  (For.  88), 

COS  T)  \-^ 

=  ££^.  tan  i  (H  +  .j)  .  .  .  (K). 
cos  F  " 

N  and  h  being  known,  to  obtain  ihe  formidcB  for  the  parallax 
in  longitude  and  latitude,  we  have  only  to  replace  in  the  formu- 
las for  the  parallax  in  right  ascension  and  declination,  the  alti- 
tude I  of  the  pole  of  the  equator,  by  that  90°  —  h  of  the  pole  K 
of  the  ecliptic,  and  the  distance  i  m  of  the  star  s  from  the  merid- 
ian by  the  distance  w  c  to  the  vertical  through  the  nonagesimal. 
Let  us  change  then  in  formulae  (A),  (B),  (C),  (D),  (E),  ^F),  and 
(G),  I  into  90°  —  h,  and  q  into/c  —  fn  =  L  —  N,  L  being  the 
longitude/ c  of  the  star  s.  Besides  d  will  become  the  distance  s 
K  to  the  pole  of  the  ecliptic,  complement  of  the  latitude  X  =  sc. 
Making  these  substitutions,and  denoting  the  parallax  in  longitude 
by  n,  and  the  parallax  in  latitude  by  -r,  we  obtain  in  terms  of  the 
apparent  longitude  and  latitude. 

.    _      sin  P  sin  h    ■    ,y        at  i  tt\  /t  \ 

sm  n  = ^ sin  (L  —  N  -j-  n)  .  .  .  (L), 

sin  d 

cot  (cZ  -}-  *)  =  !!!L(J^-  ^  +  g)  (cot  d  -  sJ!l_P_^£^)  . .  (M), 
sin  (L  —  N)        ^  sin  d      ' 

tan:r=^A|jE?!j^...(N), 
sin  d 


356  APPENDIX. 

cot  (d-{-'^)=  si"(L-N+n)cos(rf+^  ^      _ 
sin  (L  —  N)  sin  (Z  cos  :r 

sin  rf  =  sin  P  cos  h  sin  (ri  +  -r)  — 
cos  {d  +  -ff)  cos  (L  —  N  +  i  n)  sin  P  sin  h 


cos  i  n 


in 


tan  A  cos  (L  —  N  +  i  n)  .„ , 

tan  y  = ^  :lj — ^  .  .  .  (a), 

cos  1  n 

sin  P  cos  h    .     ,  J   ,  ^  /t»x 

sin  ■JT  = sm  (a  +  *  —  y)  .  .  .  (K), 

cos  1/ 


and  in  terms  of  the  true  longitude  and  latitude, 

^      Psin^     .      ,T        ATN    I     /PsinA\2 

n  = sin  (L  —  N)  +   §  — : — -  I 

sin  d  y  smd  ' 

sin  (L  —  N)  cos  (L  —  N)  sin  1"  .  .  .  (S), 

P  cos/i      ■     ,,         N    ,    ,  /P  cos  AV3 


■     ,,         \    I    1  /P  cos  h\' 
sm  (c?  —  2/)  +  i  I I 

>    cos  V    r 


COS  y  ^  cos  y 

sin  2  ((Z  — 2/)  sin  1"  .  .  .  (T), 

tan  A  cos  (L  —  N  +  ^  n) 

tan  y > — —^ — '-' 

^  cos  ^  n 

To  facilitate  the  computation,  sin  n,  sin  *,  and  sin  P,  in  formu- 
lae (L),  (P),  and  (R),  may  be  replaced  by  the  arcs  themselves. 

The  distance  d  from  the  pole  of  the  ecliptic  enters  into  these 
formulae  in  place  of  the  latitude  X. 

To  find  the  apparent  distance  rf'  we  have, 
</'  =  £Z  +  *; 
for  the  apparent  latitude  X', 

X'  =  X  —  ir ; 

for  the  apparent  longitude  L', 

L'  =  L  +  n. 

The  logarithmic  formulae  o;iven  on  page  292,  were  derived 
from  equations  (L),  (O),  and  (P),  and  the  logarithmic  for- 
mula on  page  294  from  equa,  (O). 


moon's  augmented  semi-diameter.  357 

To  determine  now  the  effect  of  parallax  upon  the  apparent  di- 
ameter of  the  moon. 

Let  A  C  B,  (Fio-.  51)  represent  the  moon,  and  E  the  station  of 
an  observer  ;  also  let  R  =  apparent  semi-diameter  of  the  moon, 
and  D  =  its  distance.     The  triangle  A  E  S  gives 

sinAES=  4--'   or,  sin  R       ^^ 


ES  '       '  D 

At  any  other  distance  D'  we  should  have  for  the   apparent 
semi-diameter  R', 

AS 


sin  R'  = 

whence, 


sin_R'^   D 
sirTR      D^ 

Thus,  if  R'  =  moon's  apparent  semi-diameter  to  an  observer  at 
the  earth's  surface,  as  at  O  Fig.  (20),  R  =  the  same  as  it  would 
be  seen  from  the  centre  C,  and  S  represents  the  situation  of  the 
moon, 

sin   R'  ^  C  S  ^  sin  Z  O  S  ^  sin  Z 
sin  R       OS      sin  Z  G  S      sin  z 

But  we  have,  (see  page  351.) 

sin  Z  _  sin  {d  +  (5)      sin  {q  -j-  a) 
sin  z  sin  c^  sin  q 

or,  in   terms  of  the  apparent  longitude  and  latitude,  (see  page 

355), 

sin  Z  _  sin  {d  +  *)       sin  (L  —  N  +  n) 
sin  z  sin  d        '        sin  L  —  N) 

TT  •     -Di      sin  R  sin  (cf -f  * )  sin  (L  —  N -f  n)         ,tt\ 

Hence,      sm  R'  = ^^ — r-r-^Vi — ^-ir^ — — -  •  •  •  (U). 

sni  d  sm  (L  —  N)  ^    ^ 

Aberration  in  Longitude  and  Latitude,  and  in  Right  Ascen- 
sion and  Declination.   (Referred  to  from  Art.  114,  page  51.] 

Aberration  is  caused  by  the  motion  of  light  in  conjunction  with 
the  motion  of  the  earth.  Light  comes  to  us  from  the  sun  in  8' 
13". 2,  during  which  time  the  earth  describes  an  arc  a  =  20".36, 
of  its  orbit  ph  din  (Fig.  82,)  supposed  circular  :  p  is  the  place 
of  the  earth.     Let  us  take  any  plane  whatsoever,  which  we  will 


358  APPENDIX. 

call  relative,  passing  through  the  star,  and  let  d  d'  be  the  intersec- 
tion of  this  plane  and  the  ecliptic,  with  which  it  makes  an  angle 
k:  let  us  seek  the  quantity  9  by  which  the  aberration  displaces  the 
star  in  the  direction  perpendicular  to  this  plane.  The  question  is 
to  project  perpendicularly  to  the  relative  plane,  the  small  constant 
arc  a  which  the  earth  describes,  this  being  the  quantity  that  the 
star  is  displaced  from  its  line  of  direction,  (which  lies  in  the  rela- 
tive plane,)  in  a  direction  parallel  to  the  line  of  the  earth's  motion, 
(see  Art.  109  of  the  text) :  this  projection  is  9,  variable  according  to 
the  position  of  the  relative  plane  in  relation  to  which  it  is  esti- 
mated. The  velocity  along  the  tangent  at  p,  makes  with  p  h  an 
angle  &  —  p  c  h  =  the  arc  p  d'  ]  a  cos  6  is  then  the  projection  of 
this  velocity  on  the  line/?  A.  The  angle  of  our  two  planes  being 
k,  this  projection  will  be  reduced  to  a  cos  6  sin  k,  when  it  is  taken 
perpendicularly  to  the  relative  plane.  Thus, 
(p  =  a  sin  k  cos  6.  .  .  .     (V). 

The  aberration  displaces  the  star  from  the  relative  plane  by  this 
quantity  9,  k  designating  the  inclination  of  this  plane  to  the  eclip- 
tic, and  6  the  arc  p  d',  reckoned  from  p  the  place  of  the  earth  to  d' 
the  point  of  intersection  of  these  two  planes.  Let  us  give  to  the 
relative  plane  the  positions  which  are  met  with  in  applications. 

Let  us  suppose  at  first  that  k  =  90°,  or  sin  ^•  =  1  ;  the  relative 
plane  will  then  be  perpendicular  to  the  ecliptic.  Let  n  be  the  ver- 
nal equinox  ;  we  have  p  d'  =71  p  —  n  d' ;  n  p  is  the  longitude  of 
the  earth,  or  180°  +  that  O  of  the  sun  ;  71  d'  is  the  longitude  I  of 
the  star;  whence 

<p=  —  acos(0 — I)- 

Now,  let  M,  (Fig.  83),  be  the  true  place  of  the  star,  M'  the  star 
as  displaced  by  aberration,  K  M  is  the  circle  of  true  latitude,  K  M' 
the  circle  of  apparent  latitude,  and  M  M'  =  <p  :  this  arc  has  its  cen- 
tre C  on  the  axis  which  passes  through  the  pole  K  of  the  ecliptic  ; 
the  longitude  of  the  star  is  then  altered  by  the  part  O  O' of  the 
ecliptic  comprised  between  these  two  planes  ;  and  since  O  O'  is  to 
the  arc  M  M'  as  the  radius  1  is  to  the  radius  C  M  =  sin  K  M  = 
cos  latitude  X  of  the  star,  we  have 

aberr.  in  long.  =  —  — - —  cos  (O —  I)  ■  .  .  (W). 
cos  X 

If  the  relative  plane  is  k  c  (Fig.  84),  perpendicular  to  the  circle 


ABERRATION  IN  RIGHT  ASCENSION  AND  DECLINATION.    359 

of  latitude  K  c  d,  the  aberration  <p  perpendicularly  to  it,  will  be 
the  aberration  in  latitude.  Let  kdhe  the  ecliptic,  and o  the  earth  • 
the  angle  k  is  measured  by  the  arc  c  d  ='k]  the  arc  o  k  =  6  =  Q 
—  long,  of  k  ;  and  as  k  d  ^  90°,  long,  of  point  ^•  =  Z  —  90° ;  sub- 
stituting in  equation  (V)  we  find, 

aberr.  in  lat.  =  —  a  sin  X  sin  (O — I)  •  •  •  (X). 
These  aberrations  of  the  star  produce  a  small  apparent  orbit, 
which  is  confounded  with  its  projection  on  the  tangent  plane  to 
the  celestial  sphere.  Let  us  suppose  the  orbit  to  be  referred  to 
two  co-ordinate  axes  passing  through  the  true  place  of  the  star 
and  lying  in  the  tangent  plane,  of  which  one  is  parallel  to  the 
plane  of  the  ecliptic,  and  the  other  perpendicular  to  this,  or  tan- 
gent to  the  circle  of  latitude  at  the  star  ;    and  let =  aberr. 

cos  X 

in  long.,  and  y  =  aberr.  in  lat ;  y  will  be  the  ordinate,  and  .r  (the 
aberr.  in  long.,  reduced  to  the  parallel  through  the  star)  the  ab- 
scissa ;  we  have, 

^   = — —  cos(o— a 

cos  X  cos  X  • 

yz=  —  a  sin  X  sin  (O  —  I), 

or,  —    =  —  cos(0  —  /), 

a 


y 


=  — sin(0  —  V). 


a  sin  X 

Squaring  the  last  two  equations,   and  adding  them  together,  O 
disappears,  and  we  find 

y2  -f-  ar2  sin2  x=  a"  sin^  X  .  .  .  (Y). 

Whatever  may  be  the  place  of  the  earth,  such  is  the  equation  of 
the  apparent  orblf,  which,  as  we  perceive,  is  an  ellipse  of  which 
the  semi-axes  are  a  and  a  sin  X,  and  whose  centre  is  the  true 
place  of  the  star.  When  the  star  is  at  the  pole  of  the  ecliptic,  X  = 
90°,  and  the  ellipse  becomes  a  circle  of  which  the  radius  is  a. 
When  X  =  0,  this  ellipse  is  reduced  to  an  arc  2  a  of  the  ecliptic. 
To  find  the  aberration  in  right  ascension,  the  relative  plane 
must  be  perpendicular  to  the  equator.  Let  k  c  be  the  equator 
(Fig.  84),  p  its  pole,  p  s  d  the  relative  plane,  which  is  the  circle 
of  declination  of  the  star  s\  k  d  the  ecliptic,  o  the  earth,  k  the 


360  APPENDIX. 

vernal  equinox,  k  c  =R,s  c  =  T>.  Aberration  carries  the  star  s 
out  of  the  plane  y>  c  d  a  distance  (p,  which  it  is  the  question  to  de- 
termine.    Equa.  (V)  is  here 

9  =  a  sin  d  cos  d  o  —  a  sin  d  cos  {k  d  —  k  o) 
—  a  sin  d  (cos  k  d  cos  k  o  -\-  sin  k  d  sin  k  6) 
=  a  sin  d  cos  k  d  cos  k  o  +  a  sin  d  sin  k  d  sin  k  o, 
hutk  0  =  long,  of  earth  =  180°  +  O  ;  we  have  also  the  angle  k 
=  the  obliquity  u  of  the  ecliptic,  and  the  right  angled  spherical 
triangle  kc  d gives  by  Napier's  rules, 

cot  k  d  =  cot  R  cos  w,   sin  d  sin  k  d  =  sin  R. 

The  1st  equa.  multiplied  by  the  2d,  gives 

sin  d  cos  k  d  =  cos  R  cos  w, 

whence,        9  =  —  a  (cos  R  cos  w  cos  O  +  sin  R  sin  O). 

The  displacement  of  M  to  M'  (Fig.  83,)  conducts,  as  before,  to 
the  division  of  9  by  cos  D,  to  have  the  corresponding  arc  of  the 
equator :  thus  the  aberration  in  right  ascension  is, 

c/  R  =  —  a  sin  R  sec  D  sin  O  —  a  cos  w  cos  R  sec  D  cos  O  (Z). 

Taking  the  relative  plane  perpendicular  to  the  circle  of  decli- 
nation, we  find  for  the  aberration  in  dedinaiion, 

d  D  =  —  a  sin   D  cos  R  sin  O  —  a  cos  w  (tan.  w  cos  D.  — 
sin  R  sin  D)  cos  G  .  •  •  («)• 

These  formulae  may  easily  be  adapted  to  logarithmic  compu- 
tation : 

In  formula  (Z)  let  a  sin  R  sec  D  =  A,  and  a  cos  w  cos  R 
sec  D  -  B  ;  then, 

cos  O)  .  .  (Z'). 

=coswcotR  .  ..{b) 


dR=  —  A  (sin   O  +  ^^  cos  G) 

cos  9 

_        »     sin  G  cos  9  •{-  sin  9  cos  G 

COS  9 


d  K=  - 

-  A  (sin  G  +            cos 

Put  tan  9  =  — — 
A 

a  cos  u  cos  R  sec  D 

a  sin  R  sec  D 

and  we  shall  have 

PARALLAX   IN   RIGHT    ASCEiSSlON   AXD  DECLINATION.         361 

= sin  (O  -f  cp). 

cos  (p 

Restoring  the  value  of  A,  and  taking  — _  for  sec  D,  we  ob- 
tain, 

eiR=-_JL!HL?_.sin(0+9)  •  •  ■  (c). 
cos  D  cos  9 

The  auxiliary  arc  9  is  given  by  equation  (6) ;  it  must  be 
substituted  in  equation  (c),  with  its  sign,  and  we  then  obtain 
d  R.  tan  9,  and  the  co-efficient  of  sin  (O  +  9)  are  constant,  for 
the  same  star,  for  a  long  period  of  time,  since  these  quantities 
vary  very  slowly  with  w  and  the  precession.  Moreover,  the  co- 
efficient of  sin  (O  +  <p)  is  the  maximum  vakie  of  dl  R,  since  it, 
answers  to  sin  (O  +9)  =  1-  Thus  we  shall  be  able  to  calculate 
in  advance,  for  any  designated  star,  the  values  of  9  and  of  the 
rnaximum  of  the  aberration  in  right  ascension,  or  of  the  loga- 
rithm of  this  maximum. 

The  results   of  these  calculations  for  50  principal  stars  are 
given  in  Table  XCI,  columns  headed  M  and  9, 

If  in  equation  (a),  we  make  a  sin  D  cos  R  =  A',  and  a  cos  w 
(tan.  w  cos  D  —  sin  R  sin  D)  =  B',  we  shall  have  the  equation, 

R' 

rf  D  =  —  A'   (sin  O  +  —r-  cos  o), 

in  which  A'  and  B'  are  constants.  This  equation  is  of  the  same 
form  with  equa.  (Z').  We  therefore  have  in  the  same  manner  as 
for  the  right  ascension, 

^'   _  *  c^s  '^  ( ^^'^  "  cos  D  —  sin  R  sin  D)  _ 

tan  0  —  — : — 

A'  a  sm  D  cos  R 

_  a  sin  w  cos  D  —  a  cos  w  sin  R  sin  D  _ 
a  sin  D  cos  R 

sin  wcot  D  *      T»  /  j\ 

= — --  —  cos  6J  tan  K     .     .     (a), 

cos  R 

J  r\  A'        ■    ,^  ,   .\  a  sin  D  cos  R   , 

cos  d  cos  6 

sin  (O  +  a) .  .  .  (e). 
&  is  given  by  equation  (c?),  and  being  substituted  in  equation 
(e),   we  shall  have   <Z  D.    ^  and  the  co-efficient  of  sin  ( O  +  ^) 

46 


362  APPENDIX. 

are  constant  for  the  same  star,  and  we  can  therefore  calculate  in 
advance  the  values  of  this  arc,  and  of  the  co-efficient,  which  is 
the  maximum  of  the  aberration  in  declination.  Columns  headed 
^  and  N,  Table  XCI,  contain  the  quantities  d  and  the  loga- 
rithms of  the  maxima  of  the  aberration  in  declination  for  50  prin- 
cipal stars. 

For  convenience  in  calculation,  the  angles  9, 5,  and  the  maxima, 
M,  N,  in  Table  XCI,  have  been  rendered  positive  in  all  cases. 
This  has  been  accomplished  by  adding  12^-  to  (p  and  5  whenever 
the  calculation  conducted  to  a  negative  value,  and  by  adding  6^- 
to  O  +  <p,  or  O  +  ^,  whenever  the  co-efficient  had  the  sign  —  ;  in 
this  manner  the  sign  of  the  two  factors  is  changed,  which  does 
not  alter  the  sign  of  the  product. 

Formulmfor  the  Nutation  in  Right  Ascension  and  Declina- 
tion.    (Referred  to  from  Article  132,  p.  56). 

In  deriving  these  formulae,  we  must  begin  with  borrowing 
certain  results  established  by  Physical  Astronomy.  It  has  been 
proved,  in  confirmation  of  Bradley's  conjectures,  that  the  pheno- 
mena of  nutation  are  explicable  on  the  hypothesis  of  the  pole  of 
the  earth,  describing,  round  its  mean  place,  (that  place  which, 
see  p.  53,  it  would  hold  in  the  small  circle  described  round  the 
pole  of  the  ecliptic,  were  there  no  inequality  of  precession)  an 
ellipse,  in  a  period  equal  to  the  revolution  of  the  moon's  nodes. 
The  major  axis  of  this  ellipse  is  situated  in  the  solstitial  colure 
and  equal  to  18".50  ;  it  bears  that  proportion  to  the  minor  axis 
(such  are  the  results  of  theory)  which  the  cosine  of  the  obliquity 
bears  to  the  cosine  of  twice  the  obliquity :  consequently,  the 
minor  axis  will  be  13".77. 

Let  Cd X  represent  such  an  ellipse,  P  being  the  mean  place 
of  the  pole,  K  the  pole  of  the  ecliptic.  C  D  A  O  is  a  circle  de- 
scribed with  the  centre  P  and  radius  C  P.  T  L  is  the  ecliptic, 
T  w  the  equator,  K  P  L  the  solstitial  colure.  In  order  to  deter- 
mine the  true  place  of  the  pole,  take  the  angle  A  P  0  equal  to 
the  retrocrradation  of  the  moon's  ascending-  node  from  T  :  draw 
O  i  perpendicular  to  P  A,  and  the  point  in  the  ellipse,  through 
which  O  i  passes,  is  the  true  place  of  the  pole.  This  construc- 
tion being  admitted,  the  nutations  in  right  ascension  and  north 


NUTATION  IN  RFGHT  ASCENSION  AND  DECLINATION.       363 

polar  distance  may,  P  p  being  very  small,  be  thus  easily  com- 
puted. 

Nutation  in  North  Polar  Distance. 
Nutation  inNDP^Ptf  —  pa.  =Vr  =  Pp.  cos  /?  P  c,  nearly, 
=  P  p  cos  (A  Pp  -f  A  P  tf) 
=  Ppcos(APp  +R  — 90°) 
=  Ppsin(APp  +  R) 
R  denoting  the  right  ascension. 

Nutation  in  Right  Ascension. 

The  right  ascension  of  a  star  is,  by  the  effect  of  nutation, 
changed  from  T  iv  into  T'^5'.     Now, 

r'  t  s  =  °f'v  -\-  'T  10  +  t  s,  nearly, 
whence,  °f  w  —  T't  s  =  —  Tv  —  t  s 

=  _°|Oo^'   cos  T'T'-y  —  Pp  sin  «P  tf.^lLlf  , 

^         ^  sin  P  tf ' 

in  which  expression  T'  f  (=  T  T'  cos  T  °f'  v)  is,  as  in  the  case  of 
precession,  common  to  all  stars. 

In  order  to  reduce  farther  the  above  expression,  we  have 
pP(r=APp-fAPtf  =  APp  +R  — 90°, 

andTr  =  LZ=P«.  ^m  A Pp 
^       sin  P  K  ' 

whence,  — °f'v  —  ts  =  — Pp  sin  A  Pp  cotan  w 

—  P  p  sin  (A  P  p  +  R  —  90°),  cot  N.  P.  D 
=  —  P  jt?  sin  A  P  jo  cot  w  +  P  p  cos  ( A  P  p  +  R)  cot  S, 
S  representing  the  north  polar  distance,  and  w  the  obliquity  of  the 
ecliptic. 

But  these  forms  are  not  convenient  for  computation.  In  order 
to  render  them  convenient,  we  must,  from  the  properties  of 
the  ellipse,  deduce  the  values  of  P  p,  and  of  the  tangent  of  A  Pp, 
and  then  substitute  such  values  in  the  above  expressions: 
thus, 

P  p  _  sec  A  P  p  _  cos  AP  O  ^  cos   (12^  —  SI) 
.   PO  ~  sec  A  P  O  ~  cos  A  P  p         cos  A  P  p 


364 


APPENDIX. 


COS  SI 
= ^-p —  ,  SV  designating  the  longitude   of  the  moon's  as- 
cending node. 

Again, 

tan  A  P  p  _  p  i     ^  P  d  ^   P  d  . 
t'^rTAFO      OT      P~D       FO  ' 

hence,   tanAP/>=^.tan  APO    =  ?-^  .  tan  (123  —  U) 

=  —  :;^ —  tan  S\. 
P  O 

Now  substitute,  and  there  will  result 

The  Nutation  in  North  Polar  Distance. 

=  PQ  ^213  (sin  A  P  «  cos  R  +  cos  A  P  »  sin  R) 
cosAPp^  ^  ^  ^ 

=  P  O  (tan  A  P  p  cos  R  cos  SI  -1-  cos  SI  sin  R), 
=  —  V  d  cos  R  sin  S\  +  P  O  cos  SI  sin  R, 
=  _  6".887  cos  R  sin  SI  +  9".250  cos  S\  sin  R .  .   .  (/) 
which  is  the  difFer^nce,  as  far  as  nutation  is  concerned,  between 
the  meati  and  apparent  north  polar  distance.     The  apparent 
north  polar  distance,  therefore,  must  be  had  by  adding  the  pre- 
ceding quantity,  with  its  sign  changed,  to  the  mean. 

Nutation  in  right  ascension  =  P  c?  sin  SI  cot  u 
-f-  P  O  cos  SI  cos  R  cot  (5  +  P  cf  sin  SI  sin  R  cot  S, 
which,  as  far  as  nutation  is  concerned,  is  the  difference  of  the 
mean  and  apparent  right  ascensions  :  and,  consequently,  the 
above  expression  must  be  subtracted  from  the  mean,  in  order  to 
obtain  the  apparent  right  ascension  ;  or,  which  is  the  same,  must 
be  added  after  a  negative  sign  has  been  prefixed  ;  in  which  case, 
we  have,  substituting  for  P  O,  P  cf  their  numerical  values. 

The  Nutation  in  Right  Ascension. 
=  —  6".887  sin  SI  cot  w 
_  9".250  cos  S\  cos  R  cot  5  —  6".887  sin  S\  sin  R  cot  5  .  .  .  {g). 
Formulas  (/)  and  [g)  are  of  the  same  form  with  (Z)  and 
(a)   for  the  aberration  in  right  ascension  and  declination,  and 


NUTATION     IN    RIGHT    ASCENSION  AND  DF.CLINATION.     365 

therefore  formulas  may  be  derived  from  them  similar  to  (c) 
and  (e),  adapted  to  logarithmic  computation.  The  quantities 
corresponding  to  9,  M,  6,  N,  have  been  calculated  for  the  stars  in 
the  catalogue  of  Table  XC,  and  inserted  in  Table  XCI,  in  the 
columns  headed  9',  M',  6',  N'. 

The  Sola?-  Nutation  arises  from  like  causes  as  the  Lunar,  and 
admits  of  similar  formulae.  As  an  ellipse,  made  the  locus  of  the 
true  place  of  the  pole,  served  to  exhibit  the  effects  of  the  lunar 
nutation,  so  an  ellipse,  of  different,  and  much  smaller  dimensions, 
may  be  made  to  represent  the  path  which  the  true  pole  of  the 
equator  would,  by  reason  of  the  sun's  inequality  of  force  in  caus- 
ing precession,  describe  about  the  mean  place  of  the  pole.  Thus, 
in  Figure  86,  the  ellipse  AdC  will  serve  to  represent  the  locus 
of  the  pole,  when  A  P  =  0".500,  P  rf  =  0".545,  and  A  P  O,  instead 
of  being  =  ^,  is  equal  to  2  G;  or  twice  the  sun's  longitude,  ac- 
cording to  the  order  of  the  signs  ;  the  equations,  therefore,  for  the 
solar  nutation  in  north  polar  distance,  and  right  ascension, 
analogous  to  those  of  p.  364  will  be 

The  Solar  Nutation  in  North  Polar  Distance. 
=  _  0".500  cos  R  sin  2  O  +  0".545  sin  R  cos  2  ©.  .  .  .  {h). 

The  Solar  Nutation  in  Right  Ascension. 

=  _  0".500  sin  2  O  cot  w 
—  0".545  cos  2  o  cos  R  cot  6  —  0".500  sin  2  O  sin  R  cot  5.  .  (i). 
If  the  apparent  place  of  a  star  should  be  required  with  great 
precision,  it  would  be  necessary  to  compute  the  solar  nutations 
from  these  formulae,  and  apply  them  as  corrections  to  the  mean 
right  ascension  and  declination.  The  calculation  would  be  per- 
formed after  the  same  manner  as  for  the  lunar  nutation  ;  but  it  is 
much  abridged  by  remarking  that  the  form  of  the  equations  is 
the  same  as  that  of  the  equations  for  the  lunar  nutation,  and  that 
the  co-efficients  are  very  nearly  the  0.075  of  those  of  the  latter 
equations.  Thus  we  can  make  use  of  the  same  arcs  9',  &',  and 
log.  maxima,  M',  N',  repeat  the  calculation  for  the  lunar  nuta- 
tion, taking  2  O  instead  of  SI,  and  multiply  the  nutations  in 
riffht  ascension  and  declination  thus  obtained  bv  0.075.  The 
results  will  be  the  solar  nutations  required.   (See  Prob.XX). 


36G  APPENDIX. 

FormulcB  for  computing  the  effects  of  the  Ohlateness  of  the 
EartKs  Surface  upon  the  Apparent  Zenith  Distance  and  Azi. 
muth  of  a  tStar.     (Referred  to  from  Article  148,  page  64). 

From  the  centre  of  the  earth,  an  observer  "would  see  a  star  at 
I  (Fig.  85),  and  would  have  V  for  his  zenith  :  from  the  surface 
his  zenith  is  Z,  and  he  sees  this  star  at  B  ;  I  B  =  />  is  the  parallax 
in  altitude  ;  the  azimuth  VZI  is  changed  VZB.  If  for  a  given 
time,  we  wish  to  calculate  the  apparent  zenith  distance  B  Z,  and 
the  apparent  azimuth  VZB,  we  have  first  to  resolve  the  spheri- 
cal triangle  I  Z  P,  in  which  we  know  the  two  sides  Z  P  =  co- 
latitude  and  I P  =  co-declination,  and  the  included  hour  angle  P  ; 
the  azimuth  V  Z  I  =  A,  and  the  arc  I  Z  =  w  will  thus  be  known. 
But  from  the  earth's  surface,  the  star  is  seen  at  B  :  the  azimuth 
V  Z  B  =  A  +  a  :  the  zenith  distance  BZ  =  n  +  p,  since,  V  Z  = 
i  being  very  small,  we  have  sensibly  IB  +BZ  =  BZ.  By 
reason  of  the  want  of  sphericity  of  the  earth,  parallax  then 
increases  the  true  azimuth  and  zenith  distance  of  a  star  by 
small  quantities,  a  and  p,  which  it  is  necessary  to  calculate.  In 
the  triangle  V  I  Z  we  have, 

cos  I  V  =  cos  i  cos  71  +  sin  i  sin  n  cos  A  =  cos  n  +  k  sin  n ; 
making  cos  i  =  1,  sin  i  =  i,  and  i  cos  K  =  k.     Now,  k  L  i,  and 
a  fortiori  cos  k  =  1,  sin  k  =  k ;  whence 

cos  I V  =  cos  n  cos  k  -f  sin  n  sin  k  =  cos  {n  —  k), 
and  lY  =  n  —  k  =  n  —  i  cos  A. 

Thus  we  correct  the  calculated  arc  n  by  the  quantity  —  i  cos 
A,  to  have 

lY  =■  z  =  n  —  ?■  cos  A  .  .  .  (j). 

If  this  value  of  z  be  introduced  into  equation  (12),  page  44, 
we  shall  have  p,  and  thence  the  apparent  zenith  distance  Z  =ii 
+  p  =  B  Z. 

Afterwards,  to  obtain  I  Z  B  =  a,  or  the  parallax  in  azimuth^ 
the  triangles  Z  B  V,  Z  B  I  give, 

sin  Z  B  V  _  sin  (A  +  a)         sin  Z  B  V  _  sin  a 
sTn  i      ~  sin  {z  -\-  pj  sin  n         sin  p ' 

equating  the  values  of  sin  Z  B  V, 

sin  n  sin  a  _  sin  i  sin  (A  -f  a) 
smp      ~     sin  {z-{'p) 


SOLUTION  OF  kkpler's  probli-.m.  367 

substituting  for  sin  p  its  value  sin  H  sin  (^  +  2^)  =  ^^^  ^  ^^^  -^ 
(equa.  10,  page  43),  and  reducing, 

sin  a  sin  (A  +  a) 

sin  H  sin  i  ~       sin  n 
and  as  i  is  very  small,  sin  i  sin  (A  +  a)  does  not  differ  sensibly 
from  i  sin  A,  and  we  thus  have  in  seconds  (For.  47,  page  343), 

H?,sinAsinl"  /,,>, 

a  = : .  .  .  \k). 

sin  n 


Solution  of  Kepler'' s  Problem,  hy  which  a  Body's  Place  is  found 
in  ail  Elliptical  Orbit. 

Let  APB  be  an  ellipse,  E  the  focus  occupied  by  the  sun, 
round  which  P  the  earth  or  any  other  planet  is  supposed  to  re- 
volve. Let  the  time  and  planet's  motion  be  dated  from  the  apside 
or  aphelion  A.  The  condition  given  is  the  time  elapsed  from  the 
planet's  quitting  A ;  the  residt  sought  is  the  place  P  ;  to  be  de- 
termined either  by  finding  the  value  of  the  angle  A  E  P,  or  by 
cutting  off,  from  the  whole  ellipse,  an  area  A  E  P  bearing  the 
same  proportion  to  the  area  of  the  ellipse  which  the  given  time 
bears  to  the  periodic  time. 

There  are  some  technical  terms  used  in  this  problem  which  we 
will  now  explain. 

Let  a  circle  A  M  B  be  described  on  A  B  as  its  diameter,  and  sup- 
pose a  point  to  describe  this  circle  uniformly,  and  the  whole  of  it,  in 
the  same  time  as  the  planet  describes  the  ellipse ;  let  also  t  de- 
note the  time  elapsed  during  P's  motion  from  A  to  P ;  then  if  A  M  = 

-J. X  2  A  M  B,  M  will  be  the  place  of  the  point  that  moves 

period 

uniformly,  whilst  P  is  that  of  the  planet's  ;  the  angle  A  C  M  is 

called  the  Mean  Anomaly^  and  the  angle  A  E  P  is  called  the  True 

Anomaly. 

Hence,  since  the  time  [t)  being  given,  the  angle  ACM  can 
always  be  immediately  found  (see  Art.  243,  p.  101),  we  may  vary 
the  enunciation  of  Kepler's  problem,  and  state  its  object  to  be  the 
finding  of  the  true  anomaly  in  terms  of  the  mean. 

Besides  the  mean  and  true  anomalies,  there  is  a  third  called  the 
Eccentric  Anomaly,  which  is  expounded  by  the  angle  D  C  A, 


368  APPENDIX. 

and  which  is  always  to  be  found  (geometrically)  by  producing 
the  ordinate  N  P  of  the  ellipse  to  the  circumference  of  the  circle. 
This  eccentric  anomaly  has  been  devised  by  mathematicians  for 
the  purposes  of  expediting  calculation.  It  holds  a  mean  place 
between  the  two  other  anomalies,  and  mathematically  connects 
them.  There  is  one  equation  by  which  the  mean  anomaly  is  ex- 
pressed in  terms  of  the  eccentric  ;  and  another  equation  by  which 
the  true  anomaly  is  expressed  in  terms  of  the  eccentric. 

We  will  now  deduce  the  two  equations  by  which  the  eccentric 
is    expressed,   respectively,   in   terms    of  the    ti'ue    and   mean 
anomalies. 
Let  t   =  time  of  describing,  A  P, 

"P  =  periodic  time  in  the  ellipse, 

a    =CA, 

ae=EC, 

V    =  ^  P  E  A, 

u  =  ^  D  C  A ;  (whence,  E  T,  perpendicular  to  D  T,  =  E  C 
X  sin  u), 

P    =PE, 

*   =  3.14159,  (fcc. ; 
then,  by  Kepler's  law  of  the  equable  description  of  areas, 

^^y^areaPEA^^^^areaDEA^X.p^^^P(.      '        ' 

areaofellip.  area©         -if  a^ 

T  /ET.DC  ,  AD.DCi       Fa  ,^  ^^     ■        ,  r^n     \ 
=  ——  I +  - — B  =^ — (E  C  .  sm  M  -i-  D  C .  w) 

Tgr  PI 

=  —  (e  sin  M  +  w) :  hence,  if  we  put  —  =  -, 

we  have, 

nt  =  e.smu-{-u  .  .  .  (l), 

an  equation  connecting  the  mean  anomaly  nt,  and  the  ec- 
centric u. 

In  order  to  find  the  other  equation,  that  subsists  between  the 
true  and  eccentric  anomaly,  we  must  investigate,  and  equate,  two 
values  of  the  radius  vector  p,  or  E  P. 


SOLUTION  OF  Kepler's  problem.  369 

First  value  of  p,  in  terms  of  v  the  true  anomaly, 

p  =     ^^i'^-^n     .  ..  .  (1). 
1  —  e  cos  V 

Second,  in  terms  ofii  the  eccentric  anomaly, 

p    =  a  (1   +  e  cos  u)  .  .  .  (2). 
For,  p-  =  E  N-  +  P  N2 

=  EN2  +DN2  x(l  — e^) 
=  {a  e   -\-  a  cos  ti)-  +  «^  sin^  u  (1  —  e^) 
=  a^  \e-  -{-  2  e  cos  u  +  cos^  u\  +  a- (1  —  e^ )  sin'  w 
=  a- 1  1  +  2  e  cos  «  +  e''  cos^  u\. 
Hence,  extracting  the  square  root, 

p  =  a  (1  +  e  cos  u). 

Equating  the  expressions  (1),  (2),  we  have 

(1  —  e-)  =  (1  —  e  cos  v)  {I  -\-  e  cos  ^^),  whence, 

e  +  cos  11  •         r  •       X  r 

cos  V  = ! ,  an  expression  for  v  in  terms  of  u  ; 

'  1  +  e  cos  n, 

but,  in  order  to  obtain  a  formula  fitted  to  logarithmic  computa- 
tion, we  must  find  an  expression  for  tan  -  :  now,  (see  For.  12, 
p.  341), 


V  _       //I  —  cos?^\  _       //(l  —  e)  (1  —  cos  ti)) 
2       V    Vl  4-  cos  vf  ~  V    \(i  ^e)  (1  +  cos  w)^ 


1  — e 
+ 

These  two  expressions  (l)  and  (m),  that  is, 
n  t  =  e  .  sin  u  -}-  u, 

analytically  resolve  the  problem,  and,  from  such  expressions,  by 
certain  formulae  belonging  to  the  higher  branches  of  analysis,  may 
V  be  expressed  in  the  terms  of  a  series  involvings  t. 

Instead,  however,  of  this  exact  but  operose  and  abstruse  method 
of  solution,  we  shall  now  give  an  approximate  method  of  express- 
ing the  true  anomaly  in  terms  of  the  mean, 
47 


370  APPENDIX. 

M  O  is  drawn  parallel  to  D  C.  (1.)  Find  the  half  difference  of 
the  angles  at  the  base  of  the  triangle  E  C  M,  from  this  expression, 

tan  ^  (C  E  M  —  C  M  E)  =  tan  ^  (C  E  M  +  C  M  E)  x  |— ^, 

X     "T*     C 

in  which,  C  E  M,  +  C  M  E  =  A  C  M,  the  mean  anomaly. 

(2.)  Find  C  E  M  by  adding  ^  (C  E  M  +  C  M  E)  and  ^  (C  E  M 
—  C  M  E)  and  use  this  angle  as  an  approximate  value  to  the  ec- 
centric anomaly  D  C  A,  from  which,  however,  it  really  differs  by 
L  EMO. 

(3.)  Use  this  approximate  value  ofZ.DCA  =  Z.ECTin 
computing  E  T  which  equals  the  arc  D  ]M  ;  for,  since  (see  p.  368), 

p 

t  = X  D  E  A,  and  (the  body  being  supposed  to  revolve 

area  O 

in  the  circle  A  DM)=—? —  xACM,  areaAED  =  area  A  C 
area  O 

M,  or,  the  area  D  E  C  +  area  A  C  D  =  area  D  C  M  +  area 
A  C  D  ;  consequently  the  area  D  E  C  =  the  area  D  C  M  •>  and, 
expressing  their  values, 

ETxD^  =  DMx  DC_  ^„^  ,j,„^_  E  T  =  D  M. 

Having  then  computed  E  T  =  D  M,  find  the  sine  of  the  resulting 
arc  D  M,  which  sine  =  O  T  ;  the  difference  of  the  arc  and  sine 
(E  T  —  O  T)  gives  E  O. 

(4.)  Use  E  O  in  computing  the  angle  E  M  O.  the  real  difference, 
between  the  eccentric  anomaly  D  C  A,  and  the  Z.  M  E  C  ;  add 
the  computed  ^  E  M  O  to  ^  M  E  C,  in  order  to  obtain  Z.DCA. 
The  result,  however,  is  not  the  exact  value  oi  Z.  D  C  A,  since  /L 
Z.  E  M  O  has  been  computed  only  approximately  ;  that  is,  by  a 
process  which  commenced  by  assuming  Z.  M  E  C,  for  the  value 
of  the  Z  D  C  A. 

For  the  purpose  of  finding  the  eccentric  anomaly,  this  is  the 
entire  description  of  the  process ;  which,  if  greater  accuracy  be 
required,  must  be  repeated  ;  that  is,  from  the  last  found  value  of 
Z.  D  C  A  =  Z  E  C  T,  E  T,  E  O,  and  Z  E  M  O  must  be  again 
computed. 


NOTE    TO    PROBLEM    XIV.  371 


NOTE  TO  PROBLEM  XIV,  (Page  277.) 

Rules  for  finding  the  Moon''s  Longitude,  Latitude,  Hourly 
Motions,  Equatorial  Parallax,  and  Semi-diameter,  for  a 
given,  time,  from  the  Nautical  Almanac. 

Reduce  the  given  time  to  mean  time  at  Greenwich ;  then, 
For  the  Longitude. 

Take  from  the  Nautical  Almanac  the  calculated  longitudes 
answering  to  the  noon  and  midnight,  or  midnight  and  noon,  next 
preceding  and  next  following  the  given  time.  Commencing  with 
the  longitude  answering  to  the  first  noon  or  midnight,  subtract 
each  longitude  from  the  next  following  one :  the  three  remain- 
ders will  be  [he  first  difi'erences.  Also  subtract  each  first  differ- 
ence from  the  following  for  the  second  difi'erences,  which  will 
have  the  plus  or  minus  ngn,  according  as  the  first  differences  in- 
crease or  decrease. 

Find  the  quantity  to  be  added  to  the  second  longitude  by 
reason  of  the  first  difierences,  by  the  proportion,  I'B"^-  :  excess  of 
given  time  above  time  of  second  longitude  :  :  second  first  differ- 
ence :  fourth  term. 

"With  the  given  tire  from  noon  or  midnight  at  the  side,  take 
from  Table  XCIII  the  quantities  corresponding  to  the  minutes, 
tens  of  seconds,  and  seconds  of  the  mean  or  half  sum  of  the  two 
second  differences,  at  the  top  :  the  sum  of  these  will  be  the  cor- 
rection for  second  differences,  which  must  have  the  same  sign 
as  the  mea'.i. 

The  sum  of  the  second  longitude,  the  fourth  term,  and  the  cor- 
rection for  second  differences,  will  be  the  longitude  required. 

For  the  Latitude. 

Prefix  to  north  latitudes  the  positive  sign,  but  to  south  latitudes 
the  negative  sign,  and  proceed  according  to  the  rules  for  the  lon- 
gitude, only  that  attention  must  now  be  paid  to  the  signs  of  the 
first  difierences,  which  may  either  be  flus  or  minus. 

The  sign  of  the  resulting  latitude  will  ascertain  whelhsr  it  is 
north  or  south. 


372  APPENDIX. 

For  the  Hourly  Motion  in  Longitude. 

Solve  the  proportion,  12''-  :  given  time  from  noon  or  mid- 
nio'ht  :  :  half  sum  of  second  differences  :  a  fourth  term. 

Take  the  sum  of  the  second  first  difference,  half  the  mean  of  the 
second  differences,  with  its  sign  changed,  and  this  fourth  term, 
and  divide  it  by  12 :  the  quotient  will  be  the  required  hourly 
motion  in  longitude. 

For  the  Hourly  Motion  in  Latitude. 

With  the  given  time  from  noon  or  midnight,  the  second  first 
difference  of  latitude,  and  the  mean  of  the  second  differences,  find 
the  hourly  motion  in  latitude  in  the  same  manner  as  directed  for 
finding  the  hourly  motion  in  longitude.  When  the  hourly  mo- 
tion is  positive^  the  moon  is  tending  north  ;  and  when  it  is  nega- 
tive^ she  is  tending  south.  * 

For  the  Semi-diameter  and  Equatorial  Parallax. 

The  moon's  semi-diameter  and  equatorial  parallax  may  be 
taken  from  the  Nautical  Almanac,  with  sufficient  accuracy,  by 
simple  proportion,  the  correction  for  second  differences  being  too 
small  to  be  taken  into  account,  unless  great  precision  is  required. 

Corrections  for  Third  and  Fourth  Differences. 

When  the  moon's  longitude  and  latitude  are  required  with 
great  precision,  corrections  must  also  be  applied  for  the  third  and 
fourth  differences.  To  determine  these,  take  from  the  Almanac 
the  three  longitudes  or  latitudes  immediately  preceding  the  given 
time,  and  the  three  immediately  following  it,  and  find  the  first, 
second,  third,  and  fourth  differences,  subtracting  always  each 
number  from  the  following  one,  and  paying  attention  to  the 
signs.  With  the  given  time  from  noon  or  midnight  at  the  side, 
take  from  Table  XCIV  the  quantities  answering  to  the  minutes 
and  seconds  of  the  middle  third  difference,  at  the  top.  Their  sum 
will  be  the  correction  for  third  differences,  which  must  have  the 
same  sign  as  the  middle  third  diffeience  when  the  given  time 
from  noon  or  midnight  is  less  than  6  hours ;  the  contrary  sign, 
when  the  given  time  is  more  than  6  hours. 


NOTE    TO    PROBLEM    XIV.  373 

With  the  given  time,  and  half  sum  of  fourth  differences,  take 
from  Table  XCV  the  correction  for  fourth  differences,  givino-  it 
always  the  same  sign  as  the  half  sum. 

The  sum  of  the  third  longitude  or  latitude,  the  proportional 
part  of  the  middle  first  difference  answering  to  the  given  time 
from  noon  or  midnight,  and  the  corrections  for  second,  third,  and 
fourth  differences,  having  regard  to  the  signs  of  all  the  quantities, 
will  be  the  longitude  or  latitude  required. 


ASTRONOMICAL  TABLES. 


•^ 


TABLE  I. 


Latitudes  and  Longitudes  from  the  Meridian  of  Greenwich,  of 
some  cities,  and  other  conspicuous  places. 


Names  of  Places. 

Latitude. 

Lo 

ngitude 
Degrees 

Longitude 

in 

in  Tnne. 

0 

,     „ 

o 

,     // 

b    m      s 

Albany,  Capitol, 

New  York, 

42 

39     3N 

73  44  49  W 

4  54  59.3 

Altona,  Obs. 

Denmark, 

53 

32  45  N 

9 

56  39  E 

0  39  46.6 

Baltimore,  Bat.  MonH., 

Maryland, 

39 

17  13  X 

76 

37  50  W 

5     6  31 

Berlin,  Obs. 

Germany, 

52 

31   13  N 

13 

23  52  E 

0  53  35.5 

Boston,  State  House, 

Mass'ts., 

42 

21   15  N 

71 

4     9W 

4  44  16.6 

Bremen,  Ohs. 

Germany, 

53 

4  36N 

8 

48  58  E 

0  35  15.9 

Brunswick,  BowtZom  Coll., 

Maine, 

43 

53     ON 

69 

58  51  W 

4  39  55 

Canton, 

China, 

23 

8     9N 

113 

16  54E 

7  33     8 

Cape  of  Good  Hope,  Ols. 

Africa, 

33 

56     3  S 

18 

28  45  E 

1   13  55 

Cape  Horn, 

S.  America, 

55 

58  41  S 

67 

10  53  W 

4  28  43 

Charleston,  St.  Mich's  Ch. 

S.  Carolina, 

32 

46  33  N 

79 

57  27  W 

5  19  49.8 

Charlottesville,  Univers., 

Virginia, 

38 

2     3N 

78 

31  29  W 

5  14     6 

Cincinnati,  Fort  Wash., 

Ohio, 

39 

5  54N 

84 

24     OW 

•5  37  36 

Copenhagen,  Ohs. 

Denmark, 

55 

40  53  N 

12 

34  57  E 

0  50  19.8 

Dorpat,  Obs. 

Russia, 

58 

22  47  N 

26 

43  45  E 

1  46  55 

Dublin,  Ohs. 

Ireland, 

53 

23  13  N 

6 

20  30  W 

0  25  22 

Edinburgh,  Ohs. 

Scotland, 

55 

57  20  N 

3 

10  54  W 

0  12  43.6 

Gotha,  Ohs. 

Germany, 

50 

56     5N 

10 

44     6  E 

0  42  56.4 

Gottingen,  Obs. 

Germany, 

51 

31  48  N 

9 

56  37  E 

0  39  46.5 

Greenwich,  Ohs. 

England, 

51 

28  39  N 

0 

0     0 

0     0     0 

Konigsberg,  Ohs. 

Prussia, 

54 

42  SON 

20 

30     7E 

1   22     0.5 

London,  St.  Paul's  Ch., 

England, 

51 

30  49  N 

0 

5  48  W 

0     0  23 

Marseilles,  Ohs. 

France, 

43 

17  50  N 

5 

22   15  E 

0  21  29.0 

Milan,  Ohs. 

Italy, 

45 

28     IN 

9 

11  48  E 

0  36  47.2 

Naples,  Obs. 

Italy, 

40 

51  47  N 

14 

15     4E 

0  57     0.3 

New  Haven,  College, 

Connecticut, 

41 

17  58N 

72 

57  46  W 

4  51   51.1 

New  Orleans,  City  Hall, 

Louisiana, 

29 

57  45  N 

90 

6  49  W 

6     0  27.3 

New  York,  City  Hall, 

New  York, 

40 

42  40  N 

74 

1     8W 

4  56     4.5 

Palermo,  Obs. 

Italy, 

38 

6  44N 

13 

21  24  E 

0  53  25  6 

Paramatta,  Obs. 

New  Hoi. 

33 

48  50S 

151 

1  34E 

10     4     6.3 

Paris,  Ohs. 

France, 

48 

50  13  N 

2 

20  24  E 

0     9  21.6 

Petersburgh,  Ohs. 

Russia, 

59 

56  31  N 

30 

18  57  E 

2     1   15.8 

Philadelphia,  Ind'ce.  H., 

Penn., 

39 

56  59  N 

75 

10  59  W 

5     0  44 

Point  Venus, 

Otaheite, 

17 

29  21  S 

149 

28  55  W 

9  57  56 

Princeton,  College, 

New  Jersey, 

40 

22        N 

74  35       W 

4  58  20 

Providence,  University, 

Rhode  Isl. 

41 

49  25  N 

71 

25  56 \V 

4  45  44 

Quebec,  Cast'e, 

L.  Canada, 

46 

49  12  N 

71 

16     OW 

4  45     4 

Richmond,  C  ipitol, 

Virginia, 

37 

32   17  N 

77 

27  28  W 

5     9  50 

Rome,  St.  Peter's  Ch., 

Italy, 

41 

54     8N 

12 

27     5E 

0  49  48 

Savannah,  Exchange, 

Georgia, 

32 

4  56N 

81 

7     9W 

5  24  29 

Schenectady, 

New  York, 

42 

48        N 

73 

55        W 

4  55  40 

Stockholm,  Ohs. 

Sweden, 

59 

20  31  N 

18 

3  44E 

1   13   15 

Turin,  Obs. 

Italy, 

45 

4     6N 

7 

42     6  E 

0  30  48.4 

Vienna,  Obs. 

Austria, 

48 

12  35  N 

16 

23     OE 

1     5  32 

Wardhus, 

Lapland, 

70 

22  36  N 

31 

7  54E 

2     4  32 

jWashington,  Capitol, 

Dist.  Colum. 

38 

52  54  N 

77 

1  48  W 

5     8     7.2  ( 

2  TABLE  II.     EicmtiiLs  uf  the  Planetary   Orbits. 

Epoch  forVesta,.Iiin(), Ceres, an  J  Pallas, Jan.  1,1 820,  mean  noon  at  Green- 
wich ;  for  the  oilier  planets,  Jan.  1,  1801.  mean  noon  at  Greenwich. 


Planet's 
name. 


Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 


Inclination  to 
the  Ecliptic. 

Sec-Var 

O           /            " 

7     0     91 

-1-18.2 

3  23  28.5 

—   4.6 

1   51     6.2 

-   0.2 

7     8     9.0 

—  12 

13     4     9.7 

10  37  26.2 

—  44 

34  34  55.0 

1    18  51.3 

—  22.6 

2  29  35.7 

—  15.5 

0  46  28.4 

+    3.1 

Longitude  of 
ascending  node 


45  57  30.9 
74  54  12.9 

48     0     3.5 
103   13  18.2 

171  7  40.4 
80  41   24.0 

172  39  26.8 
98  26   18.9 

111   56  37.4 
72  59  35  3 


Sec  Var  P^on^i^^de  of 
"i  Perihelion 


+  70.44 
+  51.10 

+  41.67 
+  26 

-I-    25 

+  57.18 
+  51.12 
+  23.58 


74 

128 

99 

332 

249 

53 

147 

121 

11 

89 

167 


21  46.9 

43  .53.1 

31  9.9 

23  56.6 

33  24.4 

33  46  0 

7  31.5 

7  4.3 

8  34.6 

9  29.8 
31  16.1 


Sec.  Var. 


+  93.22 

+  78  30 

+  103.15 

+  109.71 

+  157 

+  202 

+  94.59 

+  115.68 

-f  87.44 


Planet's 
name. 

Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 


Mean  distance  \   Mean  distance 
from  Sun,  or         from  Sun  in 
Semi-a.xis.     !  miles. 


0.3870981 
0.7233316 
1.0000000 
1.5236923 
2.3678700 
2.6690090 
2.7672450 
2.7728860 
5.2027760 
9.5387861 
19.1823900 


Eccentricity 

in  Parts  of  the 

Semi-axis. 


Sec.  Variation. 


Planet's 

Mean  Long. 

name. 

at  the  Epoch. 

Mercury 
Venus 

O            '           " 

166     0  48.6 
11  33     3.0 

Earth 

100  39  13.3 

Mars 

64  22  55.5 

Vesta 

278  30     0.4 

Juno 

200   16   19.1 

Ceres 

123   16   11.9 

Pallas 

108  24  57.9 

Jupiter 
Saturn 

112   15  230 
135  20     6  5 

Uranus 

177  48  23.0 

Mean  Sidereal 

Period  in  Mean 

Solar  Davs. 


87.9692580 

224.7007869 

365.2563835 

686.9796458 

1325.7431000 

1592.6608000 

1681.3931000 

1686.5388000 

4332  5848212 

10759.2198174 

30686.8208296 


Motion  in  mean 
Longitude  in  1 
yr.  of  365  days 

O   '     " 

53  43  3.6 

224  47  29.7 

—0  14  19.5 

191  17  9.1 

97  28  53 

82  25  8 

78  8  8 

77  54  26 

30  20  31.9 

12  13  36.1 

4  17  45.1 


+  .000003866 

—  .000062711 

—  .000041630 
+  .000090176 
+  .000004009 

—  .000005830 

+  .0001593.50 

—  .000312402 

—  .000025072 

Mean   Daily 
Motion  in 
Longitude. 


4  5  32.6 

1  36     7.8 

0  59     8.3 

0  31   26.7 

0  16   179 

0  13  32  9 

0  12  50.9 

0  12  48  4 

0  4  59.3 

0  2     0.6 

0  0  42.4 


TABLE  III.     Elements  of  Moon's  Orbit.     Epoch,  Jan.  1,  1801 


Mean  inclination  of  orbit 5     8  47.9 

Mean  longitude  of  node  at  epoch 13  53  17.7 

Mean  longitude  of  perigee  at  epoch 266   10     7.5 

Mean  longitude  of  moon  at  epoch     118   17     8.3 

Mean  distance  from  earth  or  semi-axis 59r. 964350 

Eccentricity  in  parts  of  semi-axis 0.0548442 

d      h       m       s  d 

Mean  sidereal  revolution 27    7    43  11.5=27.32166142 

Mean  tropical        do 27     7     43     4.7  =  27.3  1.58242 

Mean  synodical     do 29  12     44     2.9  =  29.53058872 

Mean  anomalistic  do 27  13     18  37.4  =  27.55459950 

Mean  nodical        do 27     5       5  36.0=27  21222222 

Mean  revolution  of  nodes  ;  sider.     .   .  .  ==6798d.279;      trop.  =  6788d. 50982 
Mean  revolution  of  perigee  ;  sider.  .  .  .  =  3232d. 57534 ;  trop.  =  3231d.4751 


TABLE  IV.  3 

Diametets,  Volumes,  Masses,  <^^c.,  of  Sun,  Moon,  and  Planets. 


Apparent  Diameter. 

True 

Volume. 

Least. 

at  Mean 
Distance. 

Greatest. 

Diameter. 

Mercury 

5.0 

6.9 

12.0 

0.398 

0.063 

Venus 

9.6 

16.9 

61.2 

0.975 

0.927 

Earth 

1.000 

1.000 

Mars 

3.6 

6.3 

18.3 

0.517 

0.139 

Jupiter 

30.0 

36.7 

45.9 

10.860 

1280.900 

Saturn 

17.0 

9.982 

995.000 

Uranus 

4.0 

4.332 

80.490 

Sun 

31    31.0 

32      1.8 

32    35.6 

111  454 

1384472.000 

Moon 

29    21.9 

31      7.0 

33    31.1 

0.275 

0.020 

Mass. 

Density. 

Gravity. 

Sidereal 
Rotation. 

Light  and 
Heat. 

/(    in 

s 

Mercury- 

20258  10 

2.782 

1.03 

24     5 

28 

6.680 

Venus 

¥6isTT 

0.9434 

0.98 

23  21 

7 

1.911 

Eartli 

1 

357500 

1. 

1. 

24     0 

0 

1.000 

Mars 

2546320" 

0.931 

0.33 

24  39 

21 

.431 

Jupiter 

1 
10487 

0.25S9 

2.72 

9  55 

50 

.037 

Saturn 

1 

0.1016 

1.01 

10  29 

17 

.011 

3500.2 

Uranus 

17^13 

0.2797 

0.95 

Unknown. 

.003 

Sun 

1 

0.2543 

27.90 

25  12 

0 

Moon 

26620200 

0.615 

0.16 

27     7 

43 

TABLE  V. 

Elements  of  the  Retrograde  Motion  of  the  Planets. 


Arc  of 

Duration  of 

Elongation 

Synodical 

Planets. 

Retrogradation. 

Retrogradation. 

at  the  Stations. 

Revolution. 

O         '                 0         ' 

d     h             d      h 

O          '                   O             ' 

Mercury 

9  22  to  15  44 

23   12  to     21    12 

14  49  to  20     51 

116  days. 

Venus 

14  35  to  17  12 

40  21   to     43   12 

27  40  to  29     41 

584 

Mars 

10     6  to  19  35 

60   18  to     80  15 

128  44  to   146  37 

780 

Jupiter 
Saturn 

9  51   to     9  59 

116   18  to  122   12 

113  35  to  116  42 

399 

6  41   to     6  55 

138   18  to  135     9 

107  25  to  110  46 

378 

Uranus 

3  36 

151 

103  30 

370 

4  TABLE  VI. 

Elements  of  the  Orbits  of  the  Satellites. 

The  distances  are  expressed  in  equatorial  radii  of  the  primaries.  The 
epoch  is  Jan.  1,  1801.  The  periods,  &c.,  are  expressed  in  mean 
solar  days. 

I.     Satellites  of  Jupiter. 


Sat. 

Mean  Distance. 

Sidereal 
Revolution. 

Inclination  of 

Orbit  to  that 

of  Jupiter. 

Mass;  that 
of  Jupiter 

being 
1000000000 

1 
2 
3 

4 

6.04853 

9.62347 

15.35024 

26.99835 

d        h       m 
1      18     28 
3     13     14 
7       3     43 
16     16     32 

O              '             " 

3       5     30 

Variable. 
Variable. 
2     58     48 

17328 
23235 
88497 
42659 

II.     Satellites  of  Saturn. 


Sat. 

Mean 
Distance. 

Sidereal 
Revolution. 

Eccentricities  and  Inclinations. 

I 
2 
3 
4 
5 
6 
7 

3.351 
4.300 

5.284 

6.819 

9.524 

22.081 

64.359 

d      h       in 

0  22     38 

1  8     53 

1  21     18 

2  17     45 
4     12     25 

15     22     41 
79       7     55 

The  orbits  of  the  six  interior 
satellites    are  nearly  circular, 
and  verj"  nearly  in  the  plane  of 
the  ring.     That  of  the  seventh 
is  considerably  inclined  to  the 
rest  and  approaches  nearer  to 
coincidence  with  the  ecliptic. 

III.     Satellites  of  Uranus. 


Sat. 

Mean 
Distance 

Sidereal  Period. 

Inclination  to  Ecliptic. 

1 
2 
3 

4 
5 
6 

13.120 
17.022 
19.845 
22.752 
45.507 
91.008 

d       li        m         s 

5     21     25       0 

8     16     56       5 

10     23       4       0 

13     11       8     59 

38       1     48       0 

107     16     40       0 

Their  orbits  are  inclined 
about    78°     58'     to    the 
ecliptic,  and  their  motion 
is    retrograde.     The    pe- 
riods of  the  2d  and  4th 
require   a  trifling  correc- 
tion.    The  orbits  appear 
to  be  nearly  circles. 

TABLE  VII.     Saturn's  Ring. 


Miles. 

Extprior  diameter  of  exterior  ring =  176418 

Intefor  ditto =  155272 

Exterior  diameter  of  interior  ring =  151690 

Interior  dit'o =  117339 

Equatorial  diameter  of  the  bodv =  79 1 60 

Interval  between  the  planet  and  interior  ring  .  .   .  .   =  19090 

Interval  of  the  rings =  1791 

Thickness  of  the  rings  not  exceeding =  100 


TABLE  X.  7 

Parallax  of  the  Sini,  on  the  first  day  of  each  Month:  tlie  mean 
horizontal  Parallax  heinsi:  assumed  =■  8  .60. 


Alti- 
tude. 

Jin. 

Feb. 
Dec. 

March. 
Nov. 

April. 
Oct. 

May. 
Sept. 

June. 
Aug. 

July. 

0 

0 

8.75 

8.73 

8.67 

8.60 

8.53 

8.48 

8.46 

5 

8.73 

8.69 

8.64 

8.56 

8.50 

8.44 

8.42 

10 

8.62 

8.59 

8.54 

8.47 

8.40 

8.35 

8.33 

15 

8.45 

8.43 

8.38 

8.30 

8.24 

8.19 

8.17 

20 

8.22 

8.20 

8.15 

8.08 

8.01 

7.97 

7.95 

25 

7.93 

7.91 

7.86 

7.79 

7.73 

7.68 

7.67 

30 

7.58 

7.56 

7.51 

7.45 

7.39 

7.34 

7.33 

35 

7.17 

7.15 

7.11 

7.04 

6.99 

6.94 

6.93 

40 

6.70 

6.68 

6.64 

6.59 

6.53 

6.49 

6.48 

45 

6.19 

6.17 

6.13 

6.08 

6.03 

5.99 

5.98 

50 

5.62 

5.61 

5.58 

5.53 

5.48 

5.45 

5.44 

55 

5.02 

5.01 

4.98 

4.93 

4.89 

4.86 

4.85 

60 

4.37 

4.36 

4.34 

4.30 

4.26 

4.24 

4.23 

65 

3.70 

3.69 

3.67 

3.63 

3.60 

3.58 

3.57 

70 

2.99 

2.98 

2.97 

2.94 

2.92 

2.90 

2.89 

75 

2.26 

2.26 

2.25 

2.23 

2.21 

2.19 

2.19 

80 

1.52 

1.52 

1.51 

1.49 

1.48 

1.47 

1.47 

85 

0.76 

0.76 

0.76 

0.75 

0.74 

0.74 

0.74 

90 

0.00 

0.00 

0.00 

000 

000 

0.00 

0.00 

TABLE  XI. 

Semi-diurnal  Arcs. 


Declination. 

Lat. 

1° 

5° 

lO"" 

15° 

20° 

25o 

30o 

o 

h  m 

h 

tn 

h     m 

h     m 

h     m 

h     m 

h     m 

5 

6  0 

6 

2 

6  4 

6  5 

6  7 

6  9 

6  12 

10 

6  1 

6 

4 

6  7 

6  11 

6  15 

6  19 

6  24 

15 

6  1 

6 

5 

6  11 

6  16 

6  22 

6  29 

6  36 

20 

6  1 

6 

7 

6  15 

6  22 

6  30 

6  39 

6  49 

25 

6  2 

6 

9 

6  19 

6  29 

6  39 

6  50 

7  2 

30 

6  2 

6 

12 

6  23 

6  36 

6  49 

7  2 

7  18 

35 

6  3 

6 

14 

6  28 

6  43 

6  59 

7  16 

7  35 

40 

6  3 

6 

17 

6  34 

6  52 

7  11 

7  32 

7  56 

45 

6  4 

6 

20 

6  41 

7  2 

7  25 

7  51 

8  21 

50 

6  5 

6 

24 

6  49 

7  14 

7  43 

8  15 

8  .'54 

55 

6  6 

6 

29 

6  58 

7  30 

8  5 

8  47 

9  42 

60 

6  7 

6 

35 

7  11 

7  51 

8  36 

9  35 

12  0 

65 

6  9 

6 

43 

7  29 

8  20 

9  25 

12  0 

—  --.' 

TABLE   XII. 


Equation  of  Time,  to  convert  Apparent  Time  into  Mean  Time. 
Argiiineiit,  Mean  Longitude  of  the  Sun. 


0« 

Is 

Us 

Ills 

IVs 

V* 

o 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

0 

+  6  58.4 

—  1  29.7 

—  3  38.7 

+  1  27.0 

+  6    4.1 

4-2  49.7 

1 

6  39.7 

1  42.0 

3  .34.2 

1  40.1 

6    6.3 

2  34.5 

2 

6  20.9 

1  53.8 

3  29.1 

1  53.1 

6    8.0 

2  18.9 

3 

6    2.1 

2    5.2 

3  23.5 

2    6.0 

6    9.1 

2    28 

4 

5  43.3 

2  15.9 

3  17.3 

2  18.9 

6    9.5 

1  46  4 

5 

5  24.5 

2  26.1 

3  10.7 

2  31.7 

6    9.3 

1  29.5 

6 

5    5.7 

2  35.9 

3    3.5 

2  44.3 

6    8.5 

1  12  3 

7 

4  46.9 

2  45.0 

2  .56.0 

2  56.7 

6    7.2 

0  54  6 

8 

4  28.2 

2  .53.6 

2  47.9 

3    8.9 

6    5.2 

0  36.6 

9 

4    9.6 

3    1.8 

2  39.5 

3  20.8 

6    2.5 

+  0  18.2 

10 

351.1 

3    9.3 

2  30.5 

3  32.5 

5  59.3 

—  0    0.4 

11 

3  32.6 

3  16.3 

2  21.2 

3  43.9 

5  55.4 

0  19.5 

12 

3  14.3 

3  22  8 

2  11.5 

3  55.0 

5  51.0 

0  38.8 

13 

2  56.2 

3  28.6 

2    1.4 

4    5.8 

5  45.8 

0  58.4 

14 

2  38.3 

3  33.9 

1  51.0 

4  16.3 

5  40.1 

1  18.2 

15 

2  20.5 

3  38.6 

140.1 

4  26.5 

5  33.7 

1  38.3 

16 

2    3.0 

3  42.7 

129.0 

4  36.3 

5  26.7 

1  58.5 

17 

1  45.7 

3  46.3 

1  17.6 

4  45.7 

5  19.2 

2  19.1 

18 

1  28.6 

3  49.2 

1    5.9 

4  54.7 

5  11.1 

2  39.8 

19 

1  11.7 

3  51.5 

0  54.1 

5    3.3 

5    2.3 

3    0.7 

20 

0  55.2 

3  53.3 

0  42.0 

5  11.3 

4  53.0 

3  21.6 

21 

0  39.1 

3  54.4 

0  29.6 

5  18.9 

4  43.1 

3  42.8 

22 

0  23.3 

3  .55.0 

0  17.1 

5  26.0 

4  32.7 

4    4.0 

23 

+  0    7.8 

3  55.0 

—  0    4.4 

5  32.6 

4  21.6 

4  25  3 

24 

—  0    7  3 

3  .54.5 

+  0    8.4 

5  38.6 

4  10.1 

4  46.6 

25 

0  22.0 

3  53.3 

0  21.5 

5  44.2 

3  57.9 

5    8.1 

26 

0  36.3 

3  51.5 

0  34.5 

5  49.3 

3  45.3 

5  29.5 

27 

0  50.3 

3  49.2 

0  47.6 

5  53.9 

3  32.1 

5  51.0 

28 

1     3.8 

3  46.2 

1    0.7 

5  57.8 

3  18.5 

6  12.3 

29 

1  16.9 

3  42.8 

1  13.8 

6     1.2 

3    4.3 

6  33.7 

30 

—  1  29.7 

—  3  38.7 

+  1  27.0 

+  6    4.1 

+  2  49.7 

—  6  54.9 

TABLE  Xin. 

Secular   Variatioji  of  Equation  of  Time. 


Argnmeiit,  5 

nil's  Mean  L 

iingitnde. 

Os 

Is 

lis 

Ills 

ivs  ;  Vs 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

0 

—  3 

+    4 

+  11 

+  14 

+  13 

+  9 

3 

2 

5 

11 

14 

13 

8 

6 

1 

6 

12 

14 

12 

8 

9 

—  1 

6 

12 

15 

12 

7 

12 

0 

7 

12 

14 

12 

7 

15 

+  1 

8 

13 

14 

11 

6 

18 

2 

8 

13 

14 

11 

6 

21 

2 

9 

14 

14 

10 

5 

24 

3 

9 

14 

14 

10 

5 

27 

4 

10 

14 

14 

9 

4 

30 

+  4 

+  11 

+  14 

-r  13 

+    9 

+  4 

TABLE   XII 


Equation  of  Time,  to  convert  Apparent  Time  into  Mean  Time. 
Aigiimeiil,  Mean  Loii^iluilc  of  the  Sun. 


o 

Vis 

VIIs 

vriis 

IXs 

Xs 

Xls 

mi  11.  sec 

7IU1I.  sec. 

m'n  sfc. 

iiiin.spc. 

viin.  sec. 

-niin.  sec. 

0 

—    0  54.9 

—  15  18.9 

—  13  58.7 

—    1  30.6 

+  11  30.0 

+  14  3.1 

1 

7  IG.l 

15  27.9 

13  43.0 

1  0.2 

11  47.0 

13  56.0 

7  37.2 

15  36.1 

13  20.3 

—  0  29.8 

12  3.3 

13  48.4 

3 

7  5S.3 

15  43.7 

13  8.9 

+  0  O.G 

12  18.7 

13  40.1 

4 

8  19.1 

15  50.5 

12  50.5 

0  31.0 

12  33.4 

1331.1 

5 

8  30.8 

15  56.5 

12  31.4 

1  1.3 

12  47.2 

13  21.6 

6 

9  0.2 

16  1.8 

12  11.0 

1  31.4 

13  0.1 

13  11.4 

7 

9  20.5 

16  6.3 

11  51.1 

2  1.3 

13  12.2 

13  0.7 

S 

9  40.6 

16  9.9 

11  20.9 

2  31.0 

13  23.5 

12  49.4 

9 

10  0.3 

16  12.9 

11  7.9 

3  0.5 

13  33.9 

12  37.4 

10 

10  19.8 

16  15.1 

10  45.4 

3  29.7 

13  43.6 

12  25.0 

11 

10  38.9 

10  10.5 

10  22.0 

3  58.6 

13  52.3 

12  12.2 

12 

10  57.8 

15  17.0 

9  58.1 

4  27.1 

14  0.2 

11  58.9 

13 

11  16.2 

16  10.0 

9  33.5 

4  55.2 

14  7.3 

11  45.1 

14 

11  31.4 

16  15.4 

9  8.4 

5  22.9 

14  13.5 

11  30.9 

15 

11  52.1 

16  13.4 

8  42.6 

5  50.2 

14  18.9 

11  1G.3 

10 

12  9.5 

16  10.4 

8  10.4 

6  17.1 

14  23.4 

11  1.1 

17 

12  26.5 

10  G.7 

7  49.6 

6  43.5 

14  27.2 

10  45.6 

18 

12  42.9 

16  2.1 

7  22.5 

7  9.3 

14  30.0 

10  29.7 

19 

12  5S.9 

15  5"..0 

6  54.9 

7  34.6 

14  32. 1 

10  13.5 

20 

13  14.4 

15  50.1 

G  27.0 

7  59.3 

14  33.3 

9  56.9 

21 

13  29.5 

15  42.9 

5  58. 5 

8  23.4 

14  33.7 

9  40.1 

22 

13  44.1 

15  34.8 

5  29.7 

8  43.9 

14  33.3 

9  23.0 

23 

13  58.0 

15  25.3 

5  0.5 

9  9.8 

14  32.2 

9  5.7 

24 

14  1  1.4 

15  16.0 

4  31.0 

9  32.0 

14  30.2 

8  48.0 

25 

14  21.1 

15  5.2 

4  1.4 

9  53.5 

14  27.5 

8  30.2 

20 

14  36.3 

14  53.6 

3  31. G 

10  14.3 

14  24.0 

8  12.2  - 

27 

14  47.9 

14  41.1 

3  1.5 

10  34.4 

14  19.9 

7  54.0 

2.S 

14  5S.S 

14  27.7 

2  31.3 

10  53.8 

14  15.0 

7  35.5 

2) 

15  9.2 

14  13.6 

2  1.0 

11  12.3 

14  9.4 

7  17.0 

30 

1  —  15  IS. 9 

—  13  58.7 

-  1  30. G 

+  11  39.0 

+  14  3.1 

4-  6  58.4 

TABLE  XIIL 

Secular   Variation  of  Equation  of  Ti?ne. 
Arjiiiment,  Sun's  Mean  Longitude. 




Vis 

VIIs 

YlUs 

IXs 

Xs 

Xls 

o 

sec. 

sec. 

ser. 

sec. 

sec. 

see. 

0 

+  4 

—  2 

—10 

—15 

—15 

—  10 

3 

3 

3 

10 

15 

14 

10 

G 

3 

4 

11 

15 

14 

9 

9 

o 

4 

12 

15 

14 

8 

12 

1 

5 

12 

15 

13 

8 

15 

-[- 1 

G 

13 

15 

13 

7 

18 

0 

7 

13 

15 

12 

G 

21 

0 

7 

14 

15 

12 

5 

24 

—1 

8 

14 

15 

11 

5 

27 

2 

9 

15 

15 

11 

4 

30 

-2 

—10 

—15 

—15 

—10 

-3 

10 


TABLE  XIV. 


Periurhutions  of  Equation  of  Time. 
III. 


II. 

0 

100 

200  300 

400 

500 

600  700 

800 

900  1000 1 

SPC. 

sec. 

sec.      sec. 

sec. 

sec 

sec.    , 

sec. 

sec. 

sec. 

sec. 

0 

1.4 

0.8 

1.0   1.7 

1.7 

1.2 

0.7  1 

0.4 

0.6 

1.4 

1,4 

100 

1.2 

1.4 

1.1   1.0 

1.6 

1.8 

1.1 

0.7 

0.6 

0.7 

1,2 

200 

0.9 

1.0 

1.2   1.2 

1.2 

1.5 

1.7 

1.1 

0.5 

0,7 

0,9 

300 

0.7 

1.1 

1.1   0.9 

1.2 

1.4 

1.5 

1.6 

1.2 

0.5 

0,7 

400 

0.5 

0.6 

1.2   1.2 

0.8 

1.0 

1.6 

1.7 

1.5 

1.2 

0,5 

500 

1.0 

0.5 

0.6   1.2 

1.4 

0.8 

0.8 

1.5 

1.9 

1,5 

1.0 

GOO 

1.7 

1.0 

0.4  0.5 

1.2 

1.4 

0.9 

0.6 

1.3 

2.0 

1.7 

700 

1.9 

1.8 

1.1 

0.4 

04 

1.1 

1.6 

1.1 

0.7 

1,2 

1.9 

800 

1.2 

1.8 

1.8 

1.2 

0.4 

0.3 

1.0  ! 

1.6 

1.2 

0.7 

1.2 

900 

0.7 

1.1 

1.7 

1.8 

1.2 

0.6 

0.2  • 

0.8 

1.6 

1,3  0,7 

1000 

1.4 

0.8 

1.0 

1.7 

1.7  1 

1.2 

0.7  ' 

0.4 

0.6 

1.4  1.4 

II. 

rv. 

sec 

sec. 

sec. 

sec. 

S'fC. 

sec.      sec.   \ 

sec. 

sec. 

sec. 

sec. 

0 

0.6 

0.7 

0.5 

0.3 

0.2 

0.6 

0.7 

0.5 

02 

0,1 

0.6 

100 

0.2 

0.7 

0.6 

0.5 

0.2 

0.3 

0.6 

0.9 

0.5 

0.2 

0.2 

200 

0.2 

0.4 

0.6 

0.5 

0.4 

0.3 

0.4 

0.6 

0.5 

0.5 

0.2 

300 

0.4 

0.2 

0.5 

0.5 

0.5 

0.4 

0.4 

0.4 

0.5 

0.5 

0.4 

400 

0.5 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.4 

0,4 

0.5 

500 

0.4 

0.5 

0.5 

0.5 

0.4 

0.4 

0.3 

0.4 

0.5 

0.3 

0,4 

600 

0.3 

0.3 

0.5 

0.6 

0.4 

0.4 

0.3 

0.5 

0.7 

0,4 

0.3 

700 

0.4 

0.2 

0.3 

0.6 

0.6 

0.4 

0.2 

0.2 

0.7 

0.7 

0.4 

800 

0.6 

0.3 

0.2 

0.3 

0,7 

06 

0.3 

0.2 

0.3 

0.8 

0,6 

900 

0.8 

0.5 

0.3 

0.1 

0.4 

0.7 

0.5 

0.3 

0.1 

0.5 

0,8 

1000 

0.6 

0.7 

10.5 

0.3 

0.2 

0.6 

0.7 

0.5 

0.2 

0.1 

0,6 

II. 

V. 

see. 

sec.   1  sec. 

.9r.-. 

src. 

SPC. 

sec. 

sec. 

sec. 

sec. 

sec. 

0 

1.0 

1.0   1.1 

1.2 

1.1 

1.0 

0.7 

0.4 

0.6 

0.9 

1.0 

100 

0.9 

0.9 

0.8 

1.0 

1.3 

1.3 

1.0 

0.7 

0,4 

0.5 

0.9 

200 

0.5 

0.7 

0.7 

0.8 

1.0 

1.0 

1.1 

1.2 

'0.9 

0,3 

0.5 

300 

0.2 

0.5 

0.7 

0.7 

0.3 

1.2 

1.5 

1.5 

1.1 

0.5 

02 

400 

0.3 

0.2 

0.5 

0.7 

0.7 

0.9 

1.3 

1.4 

1.4 

1.0 

0.3 

500 

0.8 

0.3 

0.2 

0.5 

0.7 

0.7 

1.0 

1.4 

1.4 

1,4 

0.8 

GOO 

1.3 

0.7 

0.3 

0.3 

0.5 

0.7 

!  0.9 

1.1 

1.4 

1.6 

1.3 

700 

1.5 

1.1 

0.7 

03 

0.4 

0.5 

0.8 

1.0 

1,2 

1.4 

1,5 

800 

1.3 

1.3 

1.0 

0.7 

0.4 

0.4 

0.6 

0.8 

1.0 

1.2 

1.3 

900 

1.1 

1.2 

1.2 

1.0 

0.8 

!  0.6 

]  0.5 

0.6 

0.9 

1.1 

1.1 

1000 

1.0 

1.0 

1  1.1 

1.2  i  1.1 

'  1.0 

1  0.7 

1  0.4 

|0,6 

0,9 

1,0 

Moon  and  Nutation. 

sec.       sec.      sec.   1  sec.   1  sec. 

sec. 

.inc. 

src. 

src. 

sec. 

sec. 

I. 

0.5   0.8   1.0  •  1.0  0  8 

0.5 

0.2 

0.0 

00 

0.2 

0.5 

N. 

0.1   0.1  1  0.2  0.2  1  0.2 

0.2 

0.2 

0.2 

0.2 

0.1 

0,1 

Constant  3^\0 


TABLE  XV. 


IJ 


For  converting  any  given  day  into  the  decimal  part  of  a  year 
of  365  days. 


Day 

Jan. 

Feb. 

March 
.102 

April 

May 

June 

1 

.000 

.085 

.247 

.329 

.414 

2 

.003 

.088 

.164 

.249 

.331 

.416 

3 

.006 

.090 

.167 

.252 

.334 

.419 

4 

.008 

.093 

.170 

.255 

.337 

.422 

5 

.011 

.096 

.173 

.258 

.340 

.425 

6 

.014 

099 

.175 

.260 

.342 

.427 

7 

.016 

.101 

.178 

.263 

.345 

.430 

8 

.019 

.104 

.181 

.266 

.348 

.433 

9 

.022 

.107 

.184 

.268 

.351 

.436 

10 

.025 

.110 

.186 

.271 

.353 

.438 

11 

.027 

.112 

.189 

274 

.356 

.441 

12 

.030 

.115 

.192 

.277 

.359 

.444 

13 

.033 

.118 

.195 

.279 

•362 

.446 

14 

.036 

.121 

.197 

.282 

.364 

.449 

15 

.038 

.123 

.200 

.285 

.367 

.452 

16 

.041 

.126 

.203 

.288 

.370 

.455 

17 

.044 

.129 

.205 

.290 

.373 

.458 

18 

•046 

.132 

.208 

.293 

.375 

.460 

19 

.049 

.134 

.211 

.296 

.378 

.463 

20 

.052 

.137 

.214 

.299 

.381 

.466 

21 

.055 

.140 

.216 

.301 

.384 

.468 

22 

.058 

.142 

.219 

.304 

.386 

.471 

23 

.060 

.145 

.222 

.307 

.389 

.474 

24 

.063 

.148 

.225 

.310 

.392 

.477 

25 

.066 

.151 

.227  1 

.312 

.395 

.479 

26 

.068 

.153 

.230 

.315 

.397 

.4S2 

27 

.071 

.156 

.233 

.318 

.400 

.4S5 

28 

.074 

.159 

.236 

.321 

.403 

.488 

29 

.077 

.238 

.323 

.405 

.490 

30 

.079 

.241 

.326 

.408 

.493 

31 

.082 

.241 

.411 

12 


TABLE  XV.,  Continued. 


For  converting  anxj  given  day  into  the  decimal  part  of  a  year 
of  '660  days. 


Day 

1 

July 

August 

fc-qn. 

.000 

Oct. 

Nov. 

Dec. 

.493 

.581 

.748 

.833 

.CI  5 

2 

.499 

.584 

.008 

.751 

.esG 

.918 

3 

.501 

.536 

671 

.753 

.£38 

.C21 

4 

.504 

.589 

.074 

.750 

.841 

.923 

5 

.507 

.502 

.677 

.759 

.844 

.926 

6 

.510 

.505 

.679 

.762 

846 

.929 

7 

.512 

.597 

.682 

.764 

.849 

.931 

8 

.515 

.600 

.685 

.767 

.852 

.934 

9 

.518 

.603 

.688 

.770 

855 

.937 

10 

.521 

.605 

.690 

.773 

858 

.940 

II 

.523 

.608 

.693 

.775 

.860 

.942 

12 

526 

611 

.696 

.778 

.863 

.945 

13 

.529 

614 

.699 

.781 

.860 

.948 

14 

.532 

.616 

.701 

.784 

.868 

.951 

15 

.534 

.619 

.704 

.780 

.871 

.953 

16 

.537 

.622 

.707 

.789 

.874 

.953 

17 

.540 

.625 

.710 

.792 

877 

.959 

18 

.542 

.627 

.712 

.795 

879 

.962 

19 

.545 

.630 

.715 

.797 

882 

.964 

20 

.548 

.633 

.718 

.800 

885 

.967 

21 

.551 

.635 

.721 

.803 

8S8 

.970 

22 

.553 

.638 

.723 

805 

890 

.973 

23 

.556 

.641 

.726 

.808 

893 

.975 

24 

.559 

.644 

.729 

.811 

806 

.978 

25 

.562 

.647 

.731 

.814 

.899 

.981 

26 

564 

.649 

.734 

.816 

.901 

.984 

27 

.567 

.652 

.737 

.819 

.904 

.983 

28 

.570 

.655 

.740 

.8-22 

.907 

.989 

29 

.573 

.058 

.742 

.82.-) 

.910 

.992 

30 

.575 

.660 

.745 

.827 

.912 

.1)9.') 

31 

.578 

.663 

.830 

.997 

TABLE  XVI. 


13 


For  converting  time  into  decimal  parts  of  a  day. 


Hours 

Minutes 

Scconili 

h. 

m. 

m. 

s. 

s. 

1 

.041G7 

1 

.00069 

■  31 

.02153 

1 

.03301 

■  31 

.00033 

2 

.08333 

2 

.00139 

32 

.02222 

2 

.00302 

32 

.03337 

3 

.12590 

3 

.03203 

33 

.02232 

3 

.03303 

33 

.00033 

4 

.16637 

4 

.00273 

34 

.02301 

4 

.00335 

34 

.03033 

5 

.30833 

5 

.00347 

35 

.02433 

5 

.00003 

35 

.00040 

G 

.25000 

6 

.00417 

36 

.02500 

6 

.00007 

33 

.00312 

7 

.29167 

7 

.00436 

37 

.02533 

7 

.03308 

37 

.00043 

8 

.333  53 

8 

.00556 

33 

.02333 

8 

.03003 

33 

.03()'14 

9 

.37509 

9 

.00025 

39 

.02703 

9 

.0)010 

39 

.00015 

10 

.41667 

10 

.00394 

40 

.02778 

10 

.00012 

40 

.00046 

11 

.45333 

11 

.00764 

41 

.02847 

11 

.00013 

41 

.000-17 

12 

.50000 

12 

.00333 

42 

.02917 

12 

.03014 

42 

.03049 

13 

.54167 

13 

.  .00303 

43 

.02933 

13 

.00015 

43 

.00050 

14 

.53333 

14 

.00972 

44 

.03353 

14 

.00016 

44 

.03051 

15 

.62500 

15 

01042 

45 

.03125 

15 

.00017 

45 

.00052 

16 

.6H667 

16 

.01111 

46 

.03194 

16 

.033!  8 

43 

.00053 

17 

.70833 

17 

.01180 

47 

.03364 

17 

.00020 

47 

.00054 

18 

.75000 

18 

.01253 

48 

.03333 

18 

.00021 

'IS 

.00053 

19 

.79167 

10 

.01319 

49 

.03403 

19 

.00022 

40 

.00057 

20 

.83333 

20 

.01339 

50 

.03472 

20 

.00023 

53 

.030.58 

21 

.87500 

21 

014.53 

51 

.03.542 

21 

.00024 

51 

.000.59 

22 

.91667  i 

22 

.01523 

52 

.03311 

22 

.00025 

52 

.03030 

23 

.95833  ' 

23 

01597 

53 

.03380 

23 

.00027 

53 

.0P06I 

24 

i.OOOOO 

24 

.01037 

54 

.03750 

24 

.00028 

51 

.()()0r,2 

25 

.01736 

55 

.03319 

25 

.00029 

55 

.00004 

1 

25 

.01805 

56 

.03889 

26 

.00030 

56 

.00035 

27 

.01875 

57 

.03958 

27 

.00031 

57 

.00066 

28 

.01944 

53 

.04028 

28 

.00032 

58 

.00067 

29 

.02014 

59 

.04097 

29 

.00034 

59 

.00068 

30 

.02083  1 

60 

.04167 

30 

.00035 

60 

.00069 

14 


TABLE  XVII. 


For  converting  Minutes  and,  Seconds  of  a  degree,  into  the 
deciinuL  division  of  the  same. 


Minutes          1 

Seconds         1 

1 

1 

.01667  , 

31 

.51667 

1 

.00028  - 

31 

.00861 

2 

,03333 

32 

.53333 

2 

.00056  , 

32 

.00889 

3 

.05000 

33 

.55000 

3 

.00083 

33 

.00917 

4 

.0r)007  ; 

34 

.55667 

4 

.00111 

34 

.00944 

5 

.08333 

35 

.58333 

5 

.00139 

35 

.00972 

6 

.10000 

36 

.00000 

6 

.00167 

36 

.01000 

7 

.11667 

37 

.61667 

7 

.00194 

37 

.01028 

8 

.13333 

38 

.63333 

8 

.00222 

38 

.01056 

9 

.15000 

39 

.65000 

9 

.00250 

39 

.01083 

10 

.16667 

40 

.66667 

10 

.00278 

40 

.01111 

n 

.18333 

41 

.68333 

11 

.00306 

41 

.01139 

12 

.20000 

42 

.70000 

12 

.00333 

42 

.01167 

13 

.21667 

43 

.71667 

13 

.00361 

43 

.01194 

14 

.23333 

44 

.73333 

14 

.00389 

44 

.01222 

15 

.25000 

45 

.75000 

15 

.00417 

45 

.01250 

16 

.26667 

45 

.76667 

16 

.00444 

46 

.01278 

17 

.28333 

47 

.78333 

17 

.00472 

47 

.01306 

18 

.30000 

48 

.80000 

IS 

.00500 

48 

.01333 

19 

.31667 

40 

.81667 

19 

.00528 

49 

.01361 

20 

.33333 

50 

.83333 

20 

.00556 

50 

.01389 

21 

.35000 

51 

.85000 

21 

.00583 

51 

.01417 

22 

.30667 

52 

.86667 

22 

.00611 

52 

.01444 

23 

.38333 

53 

.88333 

23 

.00639 

53 

.01472 

24 

.40000 

54 

.90000 

24 

.00667 

54 

.01500 

25 

.41667 

55 

.91667 

25 

.00694 

55 

.01528 

2fi 

.43333 

56 

.93333 

26 

.00722 

56 

.01556 

27 

.45000 

57 

.95000 

27 

.00750 

57 

.01583 

28 

.46667 

58 

.96667 

28 

.00778 

58 

.01611 

29 

.48333 

59 

.98333 

29 

.00800 

59 

.01639 

30 

.50000 

60 

1.00000 

30 

.00833 

60 

.01667 

TABLE  XVIII. 

15 

Sun's  Epochs. 

Years.  1  M.  Long.  1 

Long.Peri.  | 

I   II  !  Ill 

IV  1  V  1 

N 

VI 

VII 

362 

S30 

SO'" 

9  10  37  46.9 

so   '  "  , 
J  10  0  54 

228 

279  169 

598  ,  758 

519 

989 

1S31 

9  10  23  27.4 

9  10  1  55  588 

278  793 

130  842 

573 

235 

396 

1832B. 

9  10  9  7.9 

9  10  2  57  948 

278  418 

661 

926 

627 

482 

430 

1S33 

9  10  53  56.8 

9  10  3  59  342 

280  j 

47 

194 

11 

681 

764 

464 

1834 

9  10  39  37.3 

9  10  5  0  j 

702 

279  I 

671 

725 

95 

734 

11 

498 

1835 

9  10  25  17.8 

9  10  6  2 

02 

279  i 

296 

256 

179 

788 

257 

532 

1S3SB. 

9  10  10  58.4 

9  10  7  3 

422 

278 

920 

788 

264 

842 

504 

566 

1837 

9  10  55  47.2 

9  10  8  5 

816 

280 

549 

321 

348 

895 

787 

(CO 

183S 

9  10  41  27.8 

9  10  9  6 

176 

279 

173 

852 

432 

949 

33 

6?4 

1839 

9  10  27  8.3 

9  10  10  8 

536 

279 

798 

383 

517 

3 

279 

668 

1S40B. 

9  10  12  48.8 

9  10  11  9 

896 

278 

422 

915 

601 

56 

526 

702 

1841 

9  10  57  37.7 

9  10  12  11 

290 

280 

51 

447 

085 

110 

809 

736 

1842 

9  10  43  18.2 

9  10  13  12 

650 

279 

670 

979 

770 

164 

55 

770 

1843 

9  10  28  58. 8 

9  10  14  14 

10 

279 

300 

510 

854 

218 

301 

h04 

1844B. 

9  10  14  39.3 

9  10  15  15 

370 

278 

924 

41 

938 

272 

548 

838 

1845 

9  10  59  28.2 

9  10  16  17 

704 

280 

S.'iS 

574 

23 

325 

831 

87:i 

1846 

9  10  45  8.7 

9  10  17  19 

124 

280 

177 

106 

107 

379 

77 

90'^ 

1847 

9  10  30  49.2 

9  10  IS  20 

484 

279 

802 

637 

191 

433 

324 

940 

184SB. 

9  10  16  29.8 

9  10  19  22 

844 

278 

427 

168 

276 

487 

570 

9  ■• 

1849 

9  11  1  18.6 

9  10  20  23 

238 

280 

55 

700 

360 

540 

853 

8 

1850 

9  10  46  59.2 

9  10  21  25 

598 

280 

680 

231 

41-1 

5  4 

fi9 

41 

1851 

9  10  32  39.7 

9  10  22  26 

958 

279 

304 

762 

529 

6-8 

346 

1^ 

1852B. 

9  10  18  20.2 

9  10  23  28 

319 

278 

929 

294 

613 

701 

592 

109 

1853 

9  11  3  9.1 

9  10  24  29 

713 

280 

557 

827 

697 

755 

875 

143 

1854 

9  10  48  49.6 

9  10  25  31 

73 

280 

182 

3.38 

782 

809 

121 

177 

1855 

9  10  34  30.2 

9  10  20  32 

4?3 

279 

806 

889 

866 

863 

368 

211 

185CB. 

9  10  20  10.7 

9  10  27  34 

793 

279 

430 

421 

950 

916 

614 

245 

1857 

9  11  4  59.6 

9  10  28  35 

187 

281 

60 

953 

35 

970 

897 

279 

1858 

9  10  50  40.1 

9  10  29  37 

547 

280 

684 

485 

119 

24 

144 

313 

1859 

9  10  36  20.7 

9  10  30  39 

907 

279 

308 

16 

203 

78 

390 

347 

1860B 

9  10  22  1.2 

9  10  31  40 

267 

279 

933 

547 

288 

131 

636 

381 

1861 

9  11  6  50.1 

9  10  32  42 

661 

281 

562 

80 

372 

185 

919 

415 

1862 

9  10  52  30.6 

9  10  33  43 

21 

280 

186 

612 

456 

239 

166 

449 

1863 

9  10  38  11.1 

9  10  34  45 

381 

280 

810 

143 

541 

292 

412 

483 

1864B 

9  10  23  51.7 

9  10  35  46 

741 

279 

435 

674 

625 

346 

659 

517 

1865 

9  11  8  40.5 

9  10  36  48 

135 

281 

64 

207 

709 

400 

941 

551 

1866 

9  10  54  21.1 

9  10  37  49 

495 

280 

688 

738 

794 

453 

188 

585 

1867 

9  10  40  1.6 

9  10  38  51 

855 

280 

313 

270 

878 

507 

434 

619 

1868B 

9  10  25  42.2 

9  10  39  52  215 

279 

937 

801 

962 

561 

681 

653 

1869 

9  11  10  31.0 

9  10  40  54  609 

281 

566 

334 

47 

615 

963 

687 

1870 

9  10  56  11.6 

9  10  41  56  969  '  280 

190 

865 

!  131 

668 

210 

721 

16 


TABLE  XIX. 

Sun's  Motiovs  for  Months. 


M. 


29  8 
28  42 


0  0.0 
3  18.2 
9  11.4 
8  IDS 
0.7 


2  29  41  yS.O 

3  23  10  39  0 

3  29  15  47.9 

4  28  49  57.9 

4  29  49  6.2 

5  28  24  7.8 

5  29  23  16.1 

6  28  57  20  1 

6  29  56  34.4 

7  29  30  44.2 

8  0  29  52.G 

8  29  4  54.1 

9  0  4  2.5 

9  29  3S  12.5 
10  0  37  20.7 

10  29  12  22.3 

11  0  11  30.6 


Per. 
0 

I 

II 
0 

III 
0 

IV 

0 

0 

5 

47 

85 

i:is 

45 

10 

'Ji;3 

102 

1^  ( i  •  J 

86 

10 

27 

164 

•zi.i 

87 

15 

4^ 

24!i 

401 

131 

15 

7u 

249 

405 

\Z2 

20 

59 

329 

534 

175 

20 

C2 

33: 

53S 

r,6 

26 

110 

414 

672 

220 

28 

144 

416 

676 

221 

31 

159 

496 

806 

263 

31 

163 

499 

SIO 

265 

36 

182 

580 

943 

309 

33 

216 

5S3 

948 

310 

41 

233 

665 

81 

354 

41 

268 

668 

86 

35-. 

46 

250 

74S 

215 

3t*7 

46 

284 

750 

219 

399 

51 

300 

832 

353 

443 

51 

333 

835 

357 

444 

56 

313 

915 

486 

486 

56 

347 

917 

491 

488 

0 

7 

14 

14 

21 


21  '  13 

23  '  IS 
;8  IS 
35  22 
35  23 

41  27 

42.  27 

49  31 
49  31 
56  36 
56  SO 
C3  40 
03  .  40  I 

70  45 
70  !  45 
77  1  49 
77  ■  49 


VI 

VII 

0 

0 

125 

3 

141 

G 

17S 

G 

266 

8 

302 

8 

355 

11 

391 

11 

4S0 

14 

516 

14 

569 

17 

605 

17 

694 

20 

730 

20 

819 

23 

855 

23 

90S 

25 

944 

25 

33 

28 

09 

28 

121 

31 

158 

31 

TABLE  XX. 
Sun's  Mntinns  for  Lcn;s  and  Hours. 


Hays:  ji  Long. 


3 
4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
IS 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 
31 


0  0  0.0 

0  59  S3 

1  58  10.7 

2  .57  25.0 

3  56  33.3 


5541.6 
54  ."^^CO 
53  .';S.3 
53  6.6 
52  15.0 


9  51  23  3 

10  50  31.6 

11  49  40.0 

12  4S4S.3 

13  47  56.6 

14  47  4.9 

15  46  13  3 

16  45  21.6 

17  44  29.9 

18  43  28.3 

19  42  46.0 

20  41  54.9 

21  41  3.3 

22  40  11.6 

23  39  19.9 

24  38  28.2 

25  37  36.6 

26  .36  44.9 

27  35  53.2 

28  .35  1.6 

29  34  9.9 


Per.] 
// 
0 

I 

0 

0 

34 

0 

68 

0 

101 

1 

135 

169 

203 

236 

270 

304 

2 

338 

2 

.371 

2 

405 

2 

439 

2 

473 

2 

500 

3 

540 

3 

574 

3 

603 

3 

641  1 

3 

675 

4 

709 

4 

743 

4 

777 

4 

810 

4 

844 

4 

878 

5 

912 

5 

945 

5 

979 

5 

13 

II 

III 

IV 

^'  1 

0 

0 

0 

0 

3 

4 

1 

0 

5 

9 

3 

0  ' 

8 

13   4 

1 

11 

18 

G 

1 

14 

22 

7 

1 

16 

27 

9 

1 

19 

31 

10 

2 

22 

36 

12 

2 

25 

40 

13 

2 

27 

44 

15 

2 

30 

49 

16 

2 

33 

53 

17 

3 

36 

58 

19 

3 

38 

62 

20 

3 

41 

67 

22 

3 

44 

71 

23 

4 

47 

76 

25 

4 

49 

80 

26 

4 

52 

85 

28 

4 

55 

89 

29 

5 

58 

93 

31 

5 

60 

98 

32 

5 

63 

102 

33 

5 

66 

107 

35 

5 

68 

111 

36 

6 

71 

116 

38 

6 

74 

120 

39 

6 

77 

125 

41 

6 

79 

129 

42 

7 

82 

134 

44 

7 

N 

VI 

VII 

Hrs. 

L 

Jng. 

1  . 
VI  1 

0  t 

0 

0 

1 

t> 

27.8 

1 

0  ' 

36 

0 

2 

4 

55.7 

3 

0 

73 

0 

3 

7 

23  5 

4 

0 

109 

0 

4 

9 

51.4 

6 

1 

145 

0 

5 

12 

19.2 

7 

181 

0 

6 

14  47.1 

8 

218 

7 

17 

14.9 

10 

254 

8 

19 

42.8 

11 

290 

9 

22 

10.6 

13 

1  I 

327 

10 

24 

3S.5 

14 

1 

303 

12 

27 

6.3 

16 

2 

399 

11 

23 

34.2 

17 

2 

435 

13 

32 

20 

18 

2 

472 

14 

34 

29.0 

20 

2 

508 

2 

15 

30 

57.7 

21 

2 

544 

^ 

16 

39 

256 

23 

2 

581 

2 

17 

41 

53.4 

24 

2 

617 

2 

18 

44 

21  2 

25, 

o 

6.53 

2 

19 

46 

49.1 

27 

3 

690 

o 

20 

49 

16.9 

23 

3 

726 

2 

21 

51 

44.S 

30 

3 

762 

2 

22 

54 

12.6 

31 

3 

798 

2 

23 

56 

40  5 

32 

3 

835 

2 

24 

59 

S.3 

34 

4 

871 

2 

4 

907 

2 

4 

943 

2 

4 

980 

2 

4 

16 

3 

4 

52 

3 

4 

89 

3 

II  III 


0 
0 
0 
0 

1 

1 

1 
1 

1  2 
1  2 

1 
1 
1 


TABLE   XXL  TABLE  XXIL  17 

Sun's  Motions  for  Minutes  and  Seconds.       ^^^^P-  06/jV/w%  of 
■^  the  Lciiptic. 


Min. 

Long. 

Min. 

Long. 

Sec. 

Lon. 

Sec. 

Lon. 

1 

0  2.5 

31 

1  16.4 

1 

0.0 

31 

1.3 

2 

4.9 

32 

1  18.8 

2 

0.1 

1  32 

1.3 

3 

7.4 

33 

1  21.3 

3 

0.1 

'  33 

1.4 

4 

9.9 

34 

1  23.8 

4 

0.2 

34 

1.4 

5 

12.3 

35 

1  26.2 

5 

0.2 

35 

1.4 

6 

14.8 

36 

1  28.7 

6 

0.2 

36 

1.5 

7 

17.2 

37 

1  31.2 

7 

0.3 

:  37 

1.5 

8 

19.7 

38 

1  33.6 

8 

0.3 

38 

1.6 

9 

22.2 

39 

1  36.1 

9 

0.4 

39 

1.6 

10 

24.6 

40 

1  38.6 

10 

0.4 

40 

1.6 

11 

27.1 

41 

1  41.0 

11 

0.5 

41 

1.7 

12 

29.6 

42 

1  43.5 

12 

0.5 

42 

1.7 

13 

32.0 

43 

1  46.0 

13 

0.5 

43 

1.8 

14 

34.5 

44 

1  48.4 

14 

0.6 

44 

1.8 

15 

37.0 

45 

1  50.9 

15 

0.6 

45 

1.8 

16 

39.4 

46 

1  53.3 

16 

0.7 

46 

1.9 

17 

41.9 

47 

1  55.8 

17 

0.7 

47 

1.9 

18 

44.4 

48 

1  58.8 

18 

0.7 

48 

2.0 

19 

46.8 

49 

2  0.7 

19 

O.S 

49 

2.0 

20 

49.3 

50 

2  3.2 

20 

0.8 

50 

2.0 

21 

51.7 

51 

2  5.7 

21 

0.9 

51 

2.1 

22 

54.2 

52 

2  8.1 

22 

0.9 

52 

2.1 

23 

56.7 

53 

2  10.6 

23 

0.9 

53 

2.2 

24 

59.1 

54 

2  13.1 

24 

1.0 

54 

2.2 

25 

1  1.6 

55 

2  15.5 

25 

1.0 

55 

2.3 

26 

1  4.1 

56 

2  18.0 

26 

1.1 

56 

2.3 

27 

1  6.5 

57 

2  20.5 

27 

1.1 

57 

2.3 

28 

1  9.0 

58 

2  22.9 

28 

1.1 

58 

2.4 

29 

1  11.5 

59 

2  25.4 

29 

1.2 

59 

2.4 

30 

1  13.9 

GO 

2  27.8 

30 

1.2 

60 

2.5 

Years 

23  27 

1835 

38  80 

1836 

38.35 

1837 

37.89 

1838 

37.43 

1839 

36.98 

1840 

36.52 

1841 

36.06 

1842 

35.61 

1843 

35.15 

1844 

34.69 

1845 

34.23 

1846 

33.78 

1847 

33.32 

1848 

32.86 

1849 

32.41 

1850 

31.95 

1851 

31.49 

1852 

31.04 

1853 

30.58 

1854 

30.12 

1855 

29.66 

1856 

29.21 

1857 

28  75 

1858 

28  29 

1859 

27  84 

1860 

27.38 

1861 

26  92 

1862 

26.47 

1863 

26.01 

1864 

25.55 

TABLE  XXIIL 

Sun's  Hourly  Motion. 
Argument.     Sun's  Mean  Anomaly. 


Os 

Is 

II« 

III-' 

lYs 

\s 

o 
30 

o 
0 

2  32.92 

2  32.20 

2  30.2? 

2  27.74 

2  25.32 

2  23.60 

10 

2  32.84 

2  31.67 

2  29.46 

2  26.89 

2  24.64 

2  23.26 

20 

20 

2  32.59 

2  31.02 

2  28.61 

2  26.07 

2  24.06 

2  23.05 

10 

30 

2  32.20 

2  30.28 

2  27.74 

2  25.32 

2  23.60 

2  22.99 

0 

XI» 

Xs 

IX-^ 

VIIIs 

Ylls 

Yls 

TABLE  XXIV. 
Sun's  Semi-diatneter. 
Argument.    Sun's  Mean  Anomaly. 


OS 

I« 

11-^ 

Ills 

TVs 

Vs 

O 

30 

20 

10 

0 

o 

0 

10 

20 

30 

16  17.3 
16  17.0 
16  16  2 
16  15.0 

16  15.0 
16  13.3 

16  11.2 
16  8.8 

16  8.8 
16  6.2 
16  3.4 
16  0.6 

16  0.6 
15  57.8 
15  55.1 
15  52.7 

15  52.7 
15  .50.5 
15  48.6 
15  47.0 

15  47.0 
15  45.9 
15  45.2 
15  45.0 

XL 

X* 

IXs 

VIII* 

VII« 

Vl5 

18 


TABLE  XXV. 

Equation  of  the  Suri's  Centre. 
Argument.     Sun's  Mean  Anomaly, 


0« 

b- 

lis 

III* 

Ws 

V» 

0 

s       °        '       " 

O   '   " 

o 

-  „ 

0 

,  „ 

O    '   " 

O      '      " 

0 

11  29  59  13.9 

0  57  58.5 

40  10.7 

54  34.1 

1  38  4.8 

0  55  52.6 

1 

0   0  1  17.3 

0  59  43.9 

41  8.9 

54  30.5 

1  37  2.4 

0  54  8.7 

2 

0  3  20.6 

1  1  28.0 

42  5.1 

54  24.8 

1  35  58. 1 

0  52  24.0 

3 

0  5  23  9 

1  3  10.9 

42  59.3 

54  17.0 

1  34  52.2 

0  50  38.2 

4 

0  7  27.0 

1  4  52.6 

43  51.8 

54  7.1 

1  33  44  6 

0  48  51.6 

5 

0  9  30.0 

1  6  33.0 

44  42.1 

53  55.2 

1  32  35.4 

0  47  4.2 

6 

0  11  32.8 

1  8  12.3 

45  30.4 

53  41.0 

1  31  24.4 

0  45  16.0 

7 

0  13  35  4 

1  9  50.1 

46  16.8 

53  24  9 

1  30  11.9 

0  43  26.9 

8 

0  15  37.7 

1  11  26.5 

47  1.2 

53  6.7 

1  28  57.7 

0  41  37.0 

9 

0  17  39.6 

1  13  1.7 

47  43.5 

52  46.5 

1  27  42.0 

0  39  46.5 

10 

0  19  41.2 

1  14  35.3 

48  23.9 

52  24.2 

1  20  24.8 

0  37  55.3 

11 

0  21  42.4 

1  16  7.5 

49  2.2 

1 

51  59.8 

1  25  5.9 

0  36  3.3 

12 

0  23  43.1 

1  17  38.2 

49  38.4 

51  33.4 

1  23  45,7 

0  3410.8 

13 

0  25  43.4 

1  19  7.5 

50  12.6 

51  5.0 

1  22  23.8 

0  32  17.7 

14 

0  27  43  2 

1  20  35.2 

30  44.7 

50  34.5 

I  21  0.6 

0  30  23.8 

15 

0  29  42.3 

1  22  1.5 

51  14.9 

50  2.2 

1  19  36.0 

0  28  29.6 

16 

0  31  40.9 

1  23  26.0 

51  42.9 

49  27.7 

1  18  9.9 

0  26  34.8 

17 

0  33  38.9 

1  24  48.9 

52  8.7 

48  51.3 

1  16  42.4 

0  24  39.6 

18 

0  35  36.2 

1  26  10.3 

52  32.5 

48  13.0 

1  15  13.7 

0  22  43.9 

19 

0  37  32.9 

1  27  30.0 

52  54.3 

47  32.7 

1  13  43.5 

0  20  47.9 

20 

0  39  28.8 

1  28  48.0 

53  13.9 

46  50.4 

1  12  12.1 

0  1851.4 

21 

0  41  23.9 

1  30  4.2 

53  31.4 

46  6.3 

1  10  39.3 

0  16  54.6 

22 

0  43  18.1 

1  31  18.8 

53  46.8 

45  20.3 

1  9  5.4 

0  14  57.5 

23 

0  45  11.5 

1  32  31.7 

54  0. 1 

44  32.2 

1  7  30.3 

0  13  0.1 

24 

0  47  4.0 

1  33  42.7 

54  11.2 

43  42.4 

1  5  54  0 

0  11  2.6 

25 

0  48  55.6 

1  34  52.0 

54  20.4 

42  50.7 

I  4  16.5 

0  9  4.8 

26 

0  50  46.3 

1  35  59.4 

54  27.2 

41  57.1 

1  2  37.8 

0  7  69 

27 

0  52  36.0 

1  37  5.1 

54  32.1 

41  1.7 

1  0  58.0 

0  5  87 

28 

0  54  24.6 

1  38  8.8 

54  34.9 

40  4.5 

0  59  17.3 

0  310  5 

29 

0  56  12.1 

1  39  10.8 

54  35.4 

39  5.6 

0  57  35.4 

0  1  12.2 

30 

0  57  58.5 

1  40  10.7 

54  34.1 

38  4.8 

0  55  52.6  1        1 

TABLE  XXVI. 

Secular   Variation  of  Equation  of  Sun's  Centre. 

Argument.    Sun's  Mean  Anomaly. 


Os 

I* 

II-' 

III* 

IV» 

V* 

o 

.. 

,, 

,, 

// 

„ 

„ 

0 

—  0 

—  9 

—   15 

—  17 

—  15 

—  8 

2 

1 

9 

15 

17 

14 

8 

4 

1 

10 

16 

17 

14 

7 

6 

2 

10 

16 

17 

14 

7 

8 

2 

11 

16 

17 

13 

6 

10 

3 

11 

16 

17 

13 

6 

12 

4 

12 

17 

17 

12 

5 

14 

4 

12 

17 

16 

12 

5 

16 

5 

13 

17 

16 

12 

4 

18 

5 

13 

17 

16 

11 

3 

20 

6 

13 

17 

16 

11 

3 

22 

7 

14 

17 

16 

1-0 

2 

24 

7 

14 

17 

15 

10 

2 

26 

8 

15 

17 

15 

9 

1 

28 

8 

15 

17 

15 

9 

1 

30 

—  9 

—  15 

—  17 

—  15 

—  8 

-0 

TABLE  XXV. 

Equation  of  the  Sufi's  Centre. 
Argument.     Sun's  Mean  Anomaly. 


19 


Vis 

VII« 

VIIIs 

IXs 

Xs 

XIa 

lis 

lis 

lis 

lis 

lis 

11* 

0 

O   '    " 

O   '   " 

O   '    " 

o 

,  /, 

0   '   " 

O    '  'r 

0 

29  59  13.9 

29  2  35.2 

28  20  23.0 

28 

3  53.7 

28  18  17.1 

29  0  29.3 

1 

29  57  15.6 

29  0  52.4 

28  19  22.2 

28 

3  52.3 

28  19  17.0 

29  2  15.7 

2 

29  55  17.3 

28  59  10.5 

28  18  23.3 

28 

3  52.8 

28  20  19.0 

29  4  3.2 

3 

29  53  19.1 

28  57  29.8 

28  17  26.1 

28 

3  55.6 

28  21  22.7 

29  5  51.8 

4 

29  51  20  9 

28  55  50.0 

28  16  30.7 

28 

4  0.5 

28  22  28.4 

29  7  41.5 

5 

29  49  23.0 

28  54  11.4 

28  15  37.1 

28 

4  7.4 

28  23  35.8 

29  9  32.2 

6 

29  47  25.2 

28  52  33.8 

28  14  45.4 

28 

4  16.6 

28  24  45.1 

29  11  23.8 

7 

29  45  27.7 

28  50  57.5 

28  13  55.6 

28 

4  27.7 

28  25  56.1 

29  13  16.3 

8 

29  43  30.3 

28  49  22.4 

28  13  7.5 

28 

4  41.0 

28  27  9.0 

29  15  9.7 

9 

29  41  33.2 

28  47  48.5 

28  12  21.5 

28 

4  56.4 

28  28  23.6 

29  17  3.9 

10 

29  39  36.4 

28  46  15.7 

28  11  37.4 

28 

5  13.9 

28  29  39.8 

29  18  59.0 

11 

29  37  39.9 

28  44  44.3 

28  10  55.1 

28 

5  33.5 

28  30  57.8 

29  20  54.9 

12 

29  35  43.9 

28  43  14.1 

28  10  14.8 

28 

5  55.3 

28  .32  17.5 

29  22  51.6 

13 

29  33  48.2 

28  41  45.4 

28  9  36.5 

28 

6  19.1 

28  33  38.9 

29  24  48.9 

14 

29  31  53.0 

28  40  17.9 

28  9  0.0 

28 

6  44.9 

28  35  1.8 

29  26  46.9 

15 

29  29  58.2 

28  38  51.8 

28  8  25.6 

28 

7  12.9 

28  36  26.3 

29  28  45.5 

16 

29  28  4.0 

28  37  27.2 

28  7  53.2 

28 

7  43.1 

28  37  .52.6 

29  30  44.6 

17 

29  26  10.1 

28  36  4.0 

28  7  22.8 

28 

8  15.2 

28  39  20.3 

29  32  44-4 

18 

29  24  17.0 

28  34  42.1 

28  6  54.4 

28 

8  49.4 

28  40  49.6 

29  34  44.7 

19 

29  22  24.5 

28  33  21.9 

28  6  28.0 

28 

9  25.6 

28  42  20.3 

29  36  45.4 

20 

29  20  32.5 

28  32  3.0 

28  6  3.6 

28  10  3.9 

28  43  52.5 

29  38  46.6 

21 

29  18  41.3 

28  30  45.8 

28  5  41.4 

28  10  44.3 

28  45  26.1 

29  40  48.2 

22 

29  16  50.8 

28  29  30.1 

28  5  21.1 

28  11  26.6 

28  47  1.3 

29  42  50.1 

23 

29  15  0.9 

28  28  15.9 

28  5  2.9 

28  12  11.0 

28  48  37.7 

29  44  52.5 

24 

29  13  11.8 

28  27  3.4 

28  4  46.8 

28  12  57.4 

28  50  15.5 

29  46  55.0 

25 

29  11  23.6 

28  25  52.4 

28  4  32.6 

28  13  45.7 

28  51  54.8 

29  48  57.8 

26 

29  9  36.2 

28  24  43.2 

28  4  20.7 

28  14  36.0 

28  53  35.2 

29  51  0.8 

27 

29  7  49.5 

28  23  35.6 

28  4  10.8 

28  15  28.5 

28  .55  16.9 

29  53  3.9 

28 

29  6  3.8 

28  22  29.7 

28  4  3.0 

28  16  22.7 

28  56  59.8 

29  55  7.2 

29 

29  4  19.1 

28  21  25.4 

28  3  57.3 

28  17  18.9 

28  58  43.9 

29  57  10.5 

30 

29  2  35.2 

28  20  23.0  28  3  53.7 

28  18  17.1 

29  0  29.3 

29  59  13.9 

TABLE  XXVL 

Secular   Variation  of  Equation  of  Sun's  Centre. 
Argument.    Sun's  Mean  Anomaly. 


Vis 

VIIs 

VIIIs 

IXs 

Xs 

XI* 

o 

>, 

>' 

>t 

// 

" 

' 

0 

+  0 

+  8 

+  15 

+  17 

+  15 

+  9 

2 

1 

9 

15 

17 

15 

8 

4 

1 

9 

15 

17 

15 

8 

6 

2 

10 

15 

17 

14 

7 

8 

2 

10 

16 

17 

14 

7 

10 

3 

11 

16 

17 

14 

6 

12 

3 

11 

16 

17 

13 

6 

14 

4 

12 

16 

17 

13 

5 

16 

5 

12 

16 

17 

12 

4 

18 

5 

12 

17 

17 

12 

4 

20 

6 

13 

17 

16 

11 

3 

22 

6 

13 

17 

16 

11 

2 

24 

7 

14 

17 

16 

10 

2 

26 

7 

14 

17 

16 

10 

1 

28 

8 

14 

17 

15 

9 

1 

30 

-f  8 

+  15 

+  17 

+  15 

+  9 

+  0 

20 


TABLE  XXVII. 


Nutations. 
Argument.     Supplement  of  the  Node,  or  N. 


Solar  Nutation. 


N. 

Long. 

R.  Asc. 

Obliq. 

N. 

Long. 

R.  Asc. 

Obliq. 
—  9.3 

Long.  Obliq. 

0 

+  0.0   +0.0 

+  9.2 

500 

—  0.0 

—  0.0 

Jan. 

/f 

ff 

10 

1.0 

1.0 

9.1 

510 

1.1 

1.0 

9.3 

1 

+  0.5 

—  0.5 

20 

2.1 

2.1 

9.1 

520 

2  2 

2.0 

9.3 

11 

0.8 

0.4 

30 

3.2 

3.0 

9.0 

530 

3.3 

'2.9 

9.2 

21 

1.1 

0.2 

40 

4.2 

4.0 

8.9 

540 

4.4 

3.9 

9.0 

31 

1.2 

—  0.1 

50 

+  5.2 

+  4.9 

+  8.7 

550 

—  5.5 

—  4.8 

—  8.9 

Feb. 

60 

6.2 

6.0 

8.5 

560 

6.5 

5.7 

8.7 

10 

1.2 

+  0.1 

70 

7.2 

6.9 

8.3 

570 

7.5 

6.6 

8.4 

20 

1.0 

0.3 

80 

S.2 

7.8 

8.1 

580 

8.5 

7.5 

8.1 

March. 

2 

12 

90 

9.1 

8.7 

7.8 

590 

9.5 

8.4 

7.8 

0.7 
+  0.3 

0.4 
0.5 

100 

+  10.0 

4-  9.4 

+  7.5 

600 

—  10.4 

—  9.1 

—  7.5 

110 

10.8 

10.3 

7.1 

610 

11.2 

9.9 

7.1 

22 

—  0.1 

0.5 

120 

11.6 

11.1 

6.7 

620 

12.0 

10.6 

6.7 

April. 

1 

11 

21 

130 

12.4 

11.7 

6.3 

630 

12.8 

11.4 

6.3 

0.5 
0.8 
1.1 

0.5 
0.2 
0.2 

140 

13.1 

12.4 

5.9 

640 

13.5 

12.0 

5.9 

150 

+  13.8 

+  13.0 

+  5.5 

650 

—  14.2 

—  12.6 

—  5.4 

160 

14.4 

13.6 

5.0 

600 

14.8 

13.2 

4.9 

May. 

1 

11 

21 

31 

170 

15.0 

14.1 

4.5 

670 

15.3 

13.8 

44 

1.2 
1.2 
1.1 
0.8 

+  0.1 

—  0.1 

0.3 

0.4 

180 
190 

15.5 
15.9 

14.5 
14.8 

4.0 
3.5 

680 
690 

15.8 
16.2 

14.2 
14.7 

3.9 
3.3 

200 

+  16.3 

+  15.1 

+  2.9 

700 

—  16.6 

—  15.0 

—  2.8 

210 

16.6 

15.4 

2.4 

710 

16.9 

15.3 

2.2 

June. 
10 
20 
30 

220 

16.9 

15.6 

1.8 

720 

17.1 

15.4 

1.6 

0.4 
^0.0 

+  0.4 

0.5 
0.5 
0.5 

230 

17.1 

15.7 

1.2 

730 

17.2 

15.7 

1.1 

240 

17.2 

15.9 

0.7 

740 

17.3 

15.9 

—  0.5 

250 

+  17.3 

+  15.9 

+  0.1 

750 

—  17.3 

—  15.9 

+  0.1 

260 

17.3 

15.9 

—  0.5 

760 

17.2 

15.9 

0.7 

July. 
10 
20 
30 

0.7 
1.0 

1.2 

0.4 

0.3 

—  0.1 

270 

17.2 

15.7 

1.1 

770 

17.1 

15.7 

1.2 

280 

17.1 

15.6 

1.6 

780 

16.9 

15.4 

1.8 

390 

16.9 

15.4 

2.2 

790 

16.6 

15.3 

2.4 

300 

+  16.6 

+  15.1 

_  2.8 

800 

_16.3 

—  15.0 

+  2.9 

Aug. 

310 

16.2 

14.8 

33 

810 

15.9 

14.7 

3.5 

9 
19 

29 

1.3 
1.2 
0.9 

+  0.0 
0.4 
0.4 

320 

158 

14.5 

3.9 

820 

15.5 

14.2 

4.0 

330 

15.3 

14.1 

4.4 

830 

15.0 

13.8 

4.5 

340 

14.8 

13.6 

4.9 

840 

14.4 

13.2 

5.0 

Sept. 

350 

+  14.2 

4-  13.0 

_  5.4 

850 

_13.8 

—  12.6 

+  5.5 

8 

0.6 

0.5 

360 

13.5 

12.4 

5.9 

860 

13.1 

12.0 

5.9 

18 

28 

+  0.2 
—  0.2 

0.5 
0.5 

370 

12.8 

11.7 

6.3 

870 

12.4 

11.4 

6.3 

380 

12.0 

11.1 

6.7 

880 

11.6 

10.6 

6.7 

Oct. 

390 

11.2 

10.3 

7.1 

890 

10.8 

9.9 

7.1 

8 

0.6 

0.5 

400 

+  10.4 

+  9.4 

_  7.5 

900 

—  10.0 

—  9.1 

+  7.5 

18 

1.0 

0.3 

410 

9.5 

8.7 

7.8 

910 

9.1 

8.4 

7.8 

28 

1.2 

0.2 

420 

8.5 

7.8 

8.1 

920 

8.2 

7.5 

8.1 

Nov. 

430 

7.5 

69 

8.4 

930 

7.2 

6.6 

8.3 

7 

1.2 

+  0.0 

440 

6.5 

6.0 

8.7 

940 

6.2 

5.7 

8.5 

17 

1.2 

0.2 

450 

+  5.5 

+  4.9 

_  8.9 

950 

—  5.2 

—  4.8 

+  8.7 

27 

1.0 

0.4 

460 

4.4 

4.0 

9.0 

960 

4.2 

3.9 

8.9 

Dec. 

470 

3.3 

3.0 

9.2 

970 

3.2 

3.9 

9.0 

7 

06 

0.5 

480 

2.2 

2.1 

9.3 

980 

2.1 

2.0 

9.1 

17 

—  02 

0.5 

490 

1.1 

1.0 

9.3 

990 

1.0 

i.o 

9.1 

27 

+  0.3 

0.5 

500 

+  0.0 

+  00 

—  9.3  llOOO 

—  0.0 

-  0.0 

+  9.2 

37 

+  0.6 

—  0.5 

TABLE   XXVIII. 


TABLE  XXIX. 


21 


Lunar  Equation,  1st  part. 
Argument  L 


Lunar  Equation,  2d  part. 
Arguments  1.  and  VI. 
I. 


1 

I  |Equa 

I 

Eqii 
7.5 

0 

7.5 

500 

10 

8.0 

510 

7.0 

20 

8.4 

520 

6.6 

30 

8.9 

530 
540 

6.1 

40 

9.4 

5.6 

50 

9.8 

550 

5.2 

60 

10.3 

560 

4.7 

70 

10.7 

570 

4.3 

80 

11.1 

580 

3.9 

90 

11.5 

590 
600 

3.5 

100 

11.9 

3.1 

110 

12.3 

610 

2.7 

120 

12.6 

620 

2.4 

130 

13.0 

630 

2.0 

140 

13.3 

640 

1.7 

150 

13.6 

650 

1.4 

160 

13.8 

660 

1.2 

170 

14.1 

670 

0.9 

180 

14.3 

680 

0.7 

190 

145 

690 

0.5 

200 

14.6 

700 

0.4 

210 

14.8 

710 

0.2 

220 

14.9 

720 

0.1 

230 

14.9 

730 

0.1 

240 

15.0 

740 

0.0 

250 

15.0 

750 

0.0 

260 

15.0 

760 

0.0 

270 

14.9 

770 

0.1 

280 

14.9 

780 

0.1 

290 

14.8 

790 

0.2 

300 

14.6 

800 

0.4 

310 

14.5 

810 

0.5 

320 

14.2 

820 

0.7 

330 

14.1 

830 

0.9 

340 

13.8 

840 

1.2 

350 

13.6 

850 

1.4 

360 

13.3 

860 

1.7 

370 

13.0 

870 

2.0 

380 

12.0 

880 

2.4 

390 

12.3 

890 

2.7 

400 

11.9 

900 

3.1 

410 

11.5 

910 

3.5 

420 

11.1 

920 

3.9 

430 

10.7 

930 

4.3 

440 

103 

940 

4.7 

450 

9.8 

950 

5.2 

460 

9.4 

960 

5.6 

470 

8.9 

970 

6.1 

480 

8.4 

980 

6.6 

490 

8.0 

990 

7.0 

500 

7.5 

1000  7.5J 

VI 

0 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

0 

1.3 

1.2 

1.2 

1.1 

1.0 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

50 

1.5 

1.5 

1.5 

1.3 

1.1 

1.0 

0.9 

1.0 

1.1 

1.1 

1.1 

100 

1.7 

1.8 

1.7 

1.4 

1.2 

1.1 

1.0 

0.9 

0.9 

0.9 

0.9 

150 

1.9 

1.9 

1.8 

1.6 

1.4 

1.3 

1.0 

0.8 

0.8 

0.8 

0.7 

200 

1.9 

2.0 

2.0 

1.7 

1.5 

1.4 

1.0 

0.8 

0.8 

0.8 

0.7 

250 

2.0 

2.0 

2.0 

1.8 

1.6 

1.5 

1.1 

0.9 

0.7 

0.7 

0.6 

300 

1.9 

1.9 

1.9 

1.9 

1.7 

1.6 

1.2 

1.0 

0.8 

0.7 

0.7 

350 

1.8 

1.9 

1.9 

1.9 

1.7 

1.6 

1.4 

1.0 

1.0 

0.9 

0.8 

400 

1.6 

1.7 

1.8 

1.9 

1.7 

1.6 

1.4 

1.2 

1.1 

1.0 

1.0 

450 

1.5 

1.5 

1.6 

1.7 

1.7 

1.7 

1.6 

1.4 

1.2 

1.2 

1.1 

500 

1.3 

1.4 

1.4 

1.5 

1.7 

1.7 

1.7 

1.5 

1.4 

1.4 

1.3 

550 

1.1 

1.2 

1.2 

1.4 

1.6 

1.7 

1.7 

1.7 

1.6 

1.5 

1.5 

600 

1.0 

1.0 

1.1 

1.2 

1.4 

1.6 

1.8 

1.8 

1.8 

1.7 

1.6 

650 

0.8 

0.9 

1.0 

1.1 

1.3 

1.5 

1.7 

1.8 

1.9 

1.9 

1.8 

700 

0.7 

0.7 

0.8 

1.1 

1.2 

1.4 

1.7 

1.9 

1.9 

1.9 

1.9 

750 

0.6 

0.6 

0.7 

1.0 

1.1 

1.3 

1.6 

1.9 

1.9 

2.0 

2.0 

800 

0.7 

0.7 

0.7 

0.9 

1.1 

1.2 

1.5 

1.8 

2.0 

1.9 

1.9 

850 

0.7 

0.8 

0.8 

0.9 

0.9 

1.1 

1.4 

1.7 

1.8 

1.8 

1.9 

900 

0.9 

0.9 

0.9 

0.9 

1.0 

1.1 

1.2 

1.5 

1.7 

1.7 

1.7 

950 

1.1 

1.0 

l.l 

1.0 

1.0 

1.0 

1.1 

1.3 

1.4 

1.6 

1.5 

0 

1.3 

1.2 

1.2 

1.1 

1.0 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

1 

VI 

500 

550 

600 

650 

700 

750 

800 

850 

900 

950 

1000 

0 

1.3 

1.4 

1.4 

1.5 

1.6 

1.6 

1.6 

1.5 

1.4 

1.4 

1.3 

50 

1.1 

1.1 

1.2 

1.3 

1.5 

1.5 

1.7 

1.6 

1.5 

1.5 

1.5 

100 

0.9 

0.9 

0.9 

1.1 

1.3 

1.5 

1.6 

1.7 

1.7 

1.7 

1.7 

150 

0.7 

0.8 

0.8 

0.9 

1.2 

1.4 

1.6 

1.9 

1.8 

1.8  1.9  1 

200 

0.7 

0.7 

0.6 

0.8 

1.1 

1.2 

1.6 

1.8 

1.8 

1.8 

1.9 

250 

0.6 

0.6 

0.7  0.7 

1.0 

1.1 

1.5 

1.7 

1.9 

1.9 

2.0 

300 

0.7 

0.7 

0.7 

0.7 

0.9 

1.0 

1.4 

1.6 

1.8 

1.9 

1.9 

350 

0.8 

0.7 

0.7 

0.8 

0.9 

1.0 

1.4 

1.6 

1.6 

1.7 

1.8 

400 

1.0 

0.9 

0.8 

0.8 

0.9 

1.0 

1.2 

1.4 

1.5 

1.6 

1.6 

450 

1.1 

1.1 

1.0 

0.9 

0.9 

0.9 

1.0 

1.2 

1.4 

1.4 

1.5 

500 

1.3 

1.2 

1.2 

1.1 

0.9 

0.9 

0.9 

1.1 

1.2 

1.2 

1.3 

550 

1.5 

1.4 

1.4 

1.2 

1.0 

0.9 

0.9 

0.9 

1.0 

1.1 

1.1 

600 

1.6 

1.6 

1.5 

1.4 

1.2 

1.0 

0.8 

0.8 

0.8 

0.9 

1.0 

650 

1.8 

1.7 

1.6 

1.6 

1.3 

1.1 

0.9 

0.8 

0.7 

0.7 

0.8 

700 

1.9 

1.8 

1.8 

1.6 

1.4 

1.2 

0.9 

0.7 

0.7 

0.7 

0.7 

750 

2.0 

1.9 

1.9 

1.7 

1.5 

1.3 

1.0 

0.7 

0.7 

0.6 

0.6 

800 

1.9 

1.8 

1.8 

1.8 

1.6 

1.4 

1.1 

0.8 

0.6 

0.7 

0.7 

8.50 

1.9 

1.8 

1.8 

1.8 

1.6 

1.5 

1.2 

0.9 

0.8 

0.8 

0.7 

900 

1.7 

1.7 

1.7 

1.7 

1.6 

1.5 

1.3 

1.1 

0.9 

0.9 

0.9 

950  1.5 

1.5  1.5 

1.6 

1.7 

1.6 

1.5 

1.3 

1.2 

1.1 

1.1 

0  1.3 

1.4  1.4 

1.5 

1.6 

1.6 

1.6 

1.5 

1.4 

1.4 

1.3 

Constant  1".3. 

23 


TABLE  XXX. 


PerturhcUions  produced  hy   Venus. 

Arguments  II  and  III. 

III. 


II. 

1  ^ 
21.6 

10 

'  20 
19.8 

30 

40 

60 

60 

70 
14.7 

80 

90 

100 
128 

110 

12.5 

120 

0 

20.8 

19.0 

17.9 

16.8 

15.9 

14.0 

13.2 

li?*? 

20 

123.1 

22.7 

21.6 

21.0 

20.1 

19.3 

18.4 

17.4 

18.4 

15.5 

14.5  13.8 

13.41 

40 

23.5 

23.2 

22.9 

22.7 

22.0 

21.1 

20.4 

19.Jj 

18.7 

17.9 

16.9 

16.1 

15.3  j 

60 

22.2 

22.5 

23.1 

22.7 

22.8 

22.5 

21.9 

21.3 

20.5 

19.9 

19.1 

18.2 

17.4 

80 

20.0 

20.7 

21.4 

21.7 

22.1 

22.3 

22,2 

22.2 

21.7 

21.3 

20.7 

19.9 

19.3 

100 

17.6 

18.6 

19.2 

19.9 

20.5 

21.0 

21.6 

21.7 

21.6 

21.6 

21.5 

21.1 

20.5 

120 

15.3 

16.0 

16.9 

17.7 

18.4 

19.2 

19.8 

20.2 

20.7 

20.8 

21.1 

21.1 

20.8 

140 

13.6 

14.2 

14.8 

15.5 

16.2 

17.0 

17.6 

18.3 

19.0 

19.4 

20.0 

20.0 

20.4 

160 

12,7 

13.2 

13.6 

14.1 

14.6 

15.0 

15.7 

16.4 

17.0 

17.3 

18.1 

18.7 

19.2 

180 

12.7 

12.9 

13.1 

13.5 

13.9 

14.0 

14.5 

14.8 

15.0 

15.8 

16.4 

16.8 

17.2 

200 

13.2 

13.2 

13.2 

13.4 

13.7 

13.8 

14.1 

14.2 

14.5 

14.5 

14.8 

15.2 

16.0 

220 

13.5 

13.6 

13.9 

14.1 

14.1 

14.1 

14.2 

14.3 

14.5 

14.6 

14.6 

14.7 

14.8 

240 

13.6 

13,8 

14.1 

14.4 

14.6 

14.8 

14.8 

14.9 

15.1 

15.1 

15.1 

14.9 

14.8 

260 

12.8 

13.3 

13.8 

14.2 

14.6 

15.0 

15.3 

15.6 

15.5 

15.5 

15.6 

15.6 

15.6 

280 

11.5 

12.3 

13.0 

13.4 

14.0 

14.6 

15.1 

15.4 

16.0 

16.2 

16.2 

16.3 

16.2 

300 

10.1 

10.9 

11.3 

12.1 

12.9 

13.7 

14.2 

14.9 

15.4 

16.0 

16.4 

16.5 

16.7 

320 

8.2 

8.8 

9.6 

10.6 

11.3 

12.0 

12.9 

13,7 

14.3 

15.0 

15.8 

16.3 

16.8 

340 

6.9 

7.5 

8.1 

8.4 

9.4 

10.1 

11.1 

11.9 

12.7 

13.6 

14,4 

15.2 

16.0 

360 

6.5 

6.5 

6.8 

7.4 

8.0 

8.4 

9.1 

9.9 

10.8 

11.5 

12.6 

13.4 

14.4 

380 

6.8 

6.5 

6.3 

6.4 

6.7 

7.0 

7.6 

8.2 

8.9 

9.6 

10.6 

11.4 

12.4 

400 

7.5 

7.1 

6.7 

6.4 

6.2 

6.4 

6.5 

6.9 

7.5 

7.9 

8.7 

9.4 

10.3 

420 

9.1 

8.4 

7.6 

7.1 

6.7 

6.5 

6.3 

6.2 

6.7 

6.8 

7.2 

7.8 

8.4 

440 

10.6 

9.8 

9.0 

8.6 

7.9 

7.2 

6.7 

6.4 

6.4 

6.4 

6.6 

6.8 

7.1 

460 

12.1 

11.5 

10.5 

9.6 

9.0 

8.5 

8.0 

7.3 

6.8 

6.6 

6.5 

6.4 

6.5 

480 

13.6 

12.8 

11.9 

11.0 

10.4 

9.6 

8.8 

8.2 

7.7 

7.2 

6.8 

6.4 

6.5 

500  15.1 

14.4 

13.4 

12.4 

11.6 

10.8 

10.1 

9.3 

8.6 

8.1 

7.5 

7.1 

6.8 

520 

16.5 

15.6 

14.8 

139 

13.1 

12.3 

11.3 

10.5 

9.7 

9.1 

8.6 

7.9 

7.4 

540 

18.1 

17.5 

16.4 

15.5 

14.5 

13.7 

12.8 

11,8 

11.1 

10.4 

9.7 

8.9 

8.2 

560 

20.4 

19.3 

18.2 

17.6 

16.5 

15.4 

14.4 

13,4 

12.7 

11.6 

10.8 

10.2 

9.2 

580 

22.8 

21.7 

20.7 

19,7 

18.4 

17.6 

16.6 

15,5 

14.3 

13,4 

12.5 

11.6 

10.6 

600 

25.2 

24.1 

23  1 

22.2 

21.2 

19.9 

18.6 

17.8 

16.6 

15.6 

14.5 

13.4 

12.6 

620 

273 

26.5 

25.6 

24.7 

235 

22.5 

21.6 

20,4 

19.0 

18.1 

16,8 

15.7 

14.7 

040 

29.0 

28.5 

27.7 

26.9 

26.2 

25.1 

24.1 

22.9 

21.8 

20.8 

19,6 

18.4 

17.2 

660 

29.8 

29.6 

29.2 

28.5 

23. 1 

27.4 

26.5 

25.6 

24.5 

23.4 

22.5 

21.2 

19.8 

680 

29.7 

29.6 

29.5 

29.5 

29.1 

28.8 

28.2 

27.6 

27.0 

26.0 

25.0 

23.8 

22.8 

700 

28.8 

29.2 

29.3 

29.5 

29.5 

29.5 

29.2 

28.8 

28.4 

27.8 

27.2 

26.4 

25.2 

720 

26.9 

27.6 

28.3 

29.0 

29.2 

29.4 

29.4 

29.3 

29.1 

28.9 

28.4 

27.9 

27.3 

740 

24.7 

25.7 

26.6 

27.3 

27.9 

28.5 

29.1 

29,0 

29.2 

29.3 

29.1 

28.8 

28.4 

760 

22.2 

23.5 

24.3 

25.3 

26,2 

27.0 

27,6 

28,3 

28.6 

23.7 

28.9 

29.1 

29.0 

780 

19.6 

21.0 

22.0 

23.2 

24,2 

25.1 

25.9 

26,7 

27.3 

27.8 

28.4 

28.5 

28.7 

800 

17.2 

18.5 

19.3 

20.9 

21.8 

22.9 

23.9 

25,0 

25.8 

26.4 

26.9 

27.6 

28.1 

820 

15.2 

15.9 

17,0 

18.4 

18.9 

20.7 

21.7 

22.8 

23.8 

24.8 

25.6 

26.2 

26.6 

840 

13.2 

14.0 

15.0 

10.0 

17.0 

18.2 

18.8 

20,3 

21.7 

22.7 

23.6 

24.5 

25.3 

860 

11.5 

12.2 

130 

13.9 

14.9 

15.9 

17.1 

18,0 

18.9 

20.3 

21.4 

22.6 

23.5 

880 

11.0 

11.2 

11.5 

12.2 

13.0 

13.7 

14.8 

15,7 

16.8 

18.1 

19.1 

20.2 

21.1 

900 

11.2 

10.2 

10.9 

11.5 

12.5 

12.1 

12.8 

13,7 

14.5 

15.5 

16.6 

17.9 

18.5 

920 

12.1 

11.6 

11.5 

11.1 

11.2 

11.3 

11.7 

12.1 

12.7 

13.4 

14.4 

15.2 

16.4 

940 

14.0 

13.3 

12.6 

12.3 

11.6 

11.5 

11.3 

11.4 

11.6 

12.0 

12.8 

13,3 

14,2 

960 

16.7 

15.6 

14.6 

13.7 

13.1 

12.5 

11.9 

11.7 

11.6 

11.4 

11.7 

12.1 

12.6 

980 

19.5 

18.3 

17.3 

16.4 

15.2 

14.2 

13.4 

12.7 

12,2 

12.0 

11.9 

11.8 

11.8 

1000 

21,6 

20.8 

19.8 
20 

19.0 
30 

17,9 

16.8 

15.9 

14.7 

14,0 

13.2 
90 

12.8 

12.5 

12.2 

. 

0 

10 

40 

50 

60 

70 

80 

00 

110 

120 

TABLE  XXX. 


sa 


Perturbations  produced  by   Venus. 

Arguments  II  and  III. 

III. 


II. 

120  130 

140 

150  160 

170 

180 

190 

200 

210 

220 

230 

240 
20.1 

0 

12.2 

12.2 

12.3 

12.4 

12.8 

13.3 

13.9 

14.7 

15.6 

16.5 

17.7 

18.8 

20 

13.4 

12.9 

12.6 

12.3 

12.2 

12.4 

12.9 

13.3 

14.0 

14.6 

15.5 

16.4 

17.3 

40 

15.3 

14.4 

14.0 

13.5 

13.0 

12.9 

12.6 

12.6 

13.1 

13.5 

14.0 

14.4 

15.4 

60 

17.4 

16.7 

16.0 

15.2 

14.5 

14.0 

13.6 

13.3 

13.2 

13.2 

13.4 

13.5 

14.1 

80 

19.3 

18.7 

17.7 

17.1 

16.4 

15.9 

15.4 

14.6 

1  14.3 

13.9 

13.8 

13.7 

13.6 

100 

20.5 

20.2 

19.5 

18.9 

18.2 

17.5 

17.1 

16.3 

15.9 

15.4 

14.8 

14.6 

14.3 

120 

20.8 

20.7 

20.4 

20.0 

19.7 

19.2 

18.5 

18.0 

17.3 

16.9 

16.5 

16.2 

15.6 

140 

20.4 

20.4 

20.2 

20.0 

20.1 

19.7 

19.5 

19.3 

18.8 

18.2 

17.7 

17.4 

17.0 

160 

19.2 

19.1 

19.4 

19.7 

19.5 

19.6 

19.3 

19.6 

19.2 

19.0 

18.7 

18.4 

18.1 

180 

17.2 

17.7 

18.5 

18.5 

18.5 

18.8 

18.4 

18.8 

19.0 

19.0 

18.9 

18.6 

18.5 

200 

16.0 

16.2 

16.6 

16.8 

17.5 

17.6 

17.7 

17.9 

18.1 

18.2 

18.3 

18.3 

18.3 

220 

14.8 

15.0 

15.3 

15.7 

16.1 

16.2 

16.6 

16.8 

17.1 

17.5 

17.1 

17.4 

17.5 

240 

14.8 

14.7 

14.8 

15.0 

15.1 

15.4 

15.7 

15.8 

16.0 

16.1 

16.1 

16.3 

16.4 

260 

15.6 

15,7 

15.3 

14.8 

15.0 

15.0 

15.1 

15.0 

15.1 

15.2 

15.2 

15.1 

15.3 

280 

16.2 

16.2 

16.2 

15.9 

15.8 

15.8 

15.5 

15.4 

15.1 

14.9 

14.8 

14.7 

15.0 

300 

16.7 

17.0 

17.1 

16.9 

16.9 

16.6 

16.5 

16.3 

15.9 

15.7 

15.2 

14.9 

14.8 

320 

16.8 

17.3 

17.5 

17.6 

17.7 

17.6 

17.5 

17.2 

17.0 

16.8 

16.5 

16.1 

15.6 

340 

16.0 

16.4 

17.2 

r,.8 

17.9 

18.1 

18.3 

18.2 

18.2 

17.9 

17.5 

17.3 

16.8 

360 

14.4 

15.2 

16.0 

16.7 

17.4 

18.1 

18.4 

18.6 

18.8 

18.8 

18.8 

18.7 

18.4 

380 

12.4 

13.4 

14.3 

15.3 

16.1 

16.9 

17.5 

18.1 

18.6 

19.1 

19.3 

19.5 

19.5 

400 

10.3 

11.2 

12.3 

13.2 

14.2 

15.1 

16.0 

16.8 

17.8 

18.4 

18.8 

19.3 

19.8 

420 

8.4 

9.2 

10.0 

11.0 

12.2 

13.0 

14.1 

15.0 

15.9 

16.9 

17.7 

18.5 

19.0 

440 

7.1 

7.6 

8.4 

9.0 

9.9 

10.9 

11.8 

12.9 

13.8 

14.9 

16.0 

16.7 

17.8 

460 

6.5 

6.8 

7.2 

7.4 

8.1 

9.0 

9.7 

10.6 

11.7 

12.6 

13.8 

14.6 

15.9 

480 

6.5 

6.5 

6.4 

6.6 

7.0 

7.5 

8.2 

8.8 

9.6 

10.4 

11.5 

12.5 

13.5 

500 

6.8 

6.7 

6.5 

6.3 

6.5 

6.6 

7.0 

7.4 

8.2 

8.6 

9.4 

10.4 

11.3 

520 

7.4 

7.0 

6.8 

6.5 

6.3 

6.1 

6.3 

6.6 

7.0 

7.5 

8.0 

8.8 

9.3 

540 

8.2 

7.6 

7.2 

6.8 

6.5 

6.3 

6.2 

6.0 

6.2 

6.5 

6.9 

7.4 

7.9 

560 

9.2 

8.6 

7.9 

7.5 

6.8 

6.6 

6.3 

6.1 

6.0 

6.1 

6.2 

6.5 

6.9 

580 

10.6 

9.8 

9.1 

8.4 

7.7 

7.3 

6.6 

6.3 

6.1 

5.9 

5.7 

5.9 

60 

600 

12.6 

11.4 

10.5 

9.5 

8.7 

8.1 

7.4 

7.0 

6.4 

6.1 

5.8 

5.5 

5o- 

G20 

14.7 

13.5 

12.4 

11.4 

10.4 

9.5 

8.7 

7.9 

7.3 

6.7 

6.2 

5.6 

5.2 

640 

17.2 

16.2 

14.9 

13.7 

12.5 

11.4 

10.4 

9.5 

8.7 

7.8 

7.0 

6.5 

5.9 

660 

19.8 

19.0 

17.6 

16.5 

15.1 

13.9 

12.8 

11.5 

10.5 

9.6 

8.6 

7.7 

6.9 

680 

22.8 

21.7 

20.4 

19.3 

18.1 

16.8 

15.7 

14.2 

13.0 

11.9 

10.7 

9.6 

8.6 

700 

25.2 

24.3 

23.3 

22.1 

20.7 

19.7 

18.5 

17.3 

160 

14.3 

13.4 

12.1 

11.0 

720 

27.3 

26.4 

25.7 

24.5 

23.7 

22.5 

21.1 

20.2 

18.8 

17.7 

16.4 

15.3 

13.9 

740 

28.4 

27.7 

27.4 

26.6 

25.9 

24.9 

24.0 

22.8 

21.5 

20.6 

19.2 

18.1 

16.8 

760 

29.0 

28.7 

28.3 

27.8 

27.3 

26.8 

25.9 

25.2 

24.3 

23.0 

21.7 

20.7 

19.7 

780 

28.7 

28.7 

28.8 

28.7 

28.3 

28.0 

27.2 

26.1 

20.1 

25.2 

24.3 

23.3 

22.2 

800 

28.1 

28.3 

28.4 

28.51 

28.5 

28.4 

28.2 

27.3 

27.3 

26.7 

25.9 

25.1 

24.4 

820 

26.6 

27.3 

27.8 

28.1 

28.3 

28.1 

28.1 

28.0 

27.9 

27.7 

27.2 

26.5 

25.9 

840 

25.3 

26.2 

26.7 

27.2 

27.5 

27.9 

28.1 

28.1 

27.9 

27.9 

27.6 

27.3 

27.2 

860 

23.5 

24.5 

25.1 

25.9 

26.6 

27.1 

27.4 

27.7 

27.9 

28.0 

27.9 

27.7 

27.5 

880 

21.1 

22.4 

23.3 

24.2 

25.1 

25.8 

26.5 

27.0 

27.3 

27.5 

27.8 

28.0 

27.7 

900 

18.5 

20.1 

21.3 

22.1  1 

23.1 

24.7 

25.0 

25.7 

26.3 

26.9 

27.3 

27.5 

27.6 

920 

16.4 

17.7 

18.4 

20.01 

21.0 

22  2 

23.0 

23.9 

24.9 

25.7 

26.2 

26.9 

27.3 

940 

14.2 

14.9 

16.1 

17.5  ! 

18.2 

19.6 

20.8 

21.9 

23.0 

23.9 

24.7 

25.7 

26.1 

960 

12.6 

13.3 

14.1 

14.4 

15.9 

17.2 

17.9 

19.5 

20.5 

21.7 

22.7 

23.9 

24.7 

980 

11.8 

12.1 

12.7 

13.3 

14.1 

14.8 

15.6 

16.8 

17.6 

19.3 

20.2 

21.4 

22.6 

1000 

12  2 

12.2 

12.3 
140 

12.4 
150 

12.8 

13.3 
170 

13.9 

14.7 

15.6 

16.5 

17.6 

18.8 

20.1 

120 

130 

160 

ISO 

190 

200 

210 

220 

230 

24) 

S4 


TABLE  XXX. 


Perturb  a  tlo7is  j^'oduced  hy   Venus. 

Arguments  II.  and  III. 

III. 


11. 

0 

240 

2.50 
21.1 

260 

270 
23.4 

1  280 

24.3 

290 
25.2 

300 

310 

320 

330 

340 

350 
27.6 

:  360 

1  " 
27.6 

20.1 

22.2 

25.8 

26.6 

27.2 

27.6 

27.7 

20 

I  17.3 

18.6 

19.7 

20.9 

21.9 

23.0 

24.2 

24.9 

25.8 

26.6 

27.0 

27.4 

27.7 

40 

15.4 

16.5 

17.3 

18.3 

I  19.4 

20.5 

21.6 

22.7 

23.7 

24.9 

25.5 

26.3 

26.9 

60 

14.1 

14.6 

15.2 

16.3 

17.2 

18.1 

18.9 

20.3 

21.2 

22.3 

23.4 

24.5 

25.3 

80 

13.6 

14.0 

14.5 

14.9 

15.5 

16.3 

17.3 

18.2 

19.0 

20.0 

21.1 

22.0 

23.1 

100 

14.3 

14.3 

14.3 

14.4 

14.6 

15.0 

15.5 

16.2 

16.9 

17.7 

18.9 

19.8 

20.8 

120 

15.6 

15.2 

14.8 

14.8 

15.0 

14.9 

15.0 

15.2 

15.9 

16.3 

17.0 

17.7 

18.5 

140 

17.0 

16.6 

1  16.4 

15.8 

i  15.5 

15.4 

15.6 

15.6 

15.5 

15.6 

16.1 

16.7 

17.1 

160 

18.1 

17.7 

117.5 

17.3 

16.9 

[16.6 

16.3 

15.9 

16.1 

16.3 

16.3 

16.2 

16.5 

ISO 

18.5 

18.5 

i  18.3 

18.1 

17.9 

1  17.6 

17.5 

17.3 

17.0 

16.9 

16.7 

16.8 

16.9 

200 

18.3 

18.4 

18.2 

18.2 

18.2 

18.2 

18.1 

18.1 

17.8 

17.7 

17.6 

17.5 

17.7 

220 

17.5 

17.6 

17.8 

17.8 

18.0 

18.0 

18.2 

18.1 

18.1 

18.3 

18.4 

18.3 

18.3 

240 

16.4 

16.5 

16.7 

16.9 

17.1 

17.3 

17.3 

17.7 

17.5 

18.0 

18.3 

18.4 

18.6 

260 

15.3 

15.5 

15.5 

15.6 

15.8 

16.1 

16.4 

16.6 

1S.8 

16.9 

17.4 

17.7 

18.2 

280 

15.0 

14.9 

14.9 

14.9 

14.9 

14.7 

15.0 

15.3 

15.5 

15.9 

16.1 

16.4 

16.8 

300 

14.8 

14.6 

14.6 

14.2 

14.0 

14.0 

13.9 

13.9 

14.2 

14.5 

14.8 

15.0 

15.5 

320  '  15.6 

15.3 

14.7 

14.5 

14.4 

13.1 

13.6 

13.4 

13.3 

13.1 

13.4 

13.6 

13.8 

340 

16.8 

16.6 

16.0 

15.5 

15.2 

14.5 

14.3 

13.7 

13.1 

13.0 

12.7 

12.6 

12.6 

360 

18.4 

17.9 

17.5 

17.0 

16.5 

15.9 

15.4 

14.9 

14.3 

13.7 

13.0 

12.6 

12.3 

380 

19.5 

19.2 

18.9 

18.5 

17.9 

17.7 

16.9 

16.4 

15.8 

15.0 

14.5 

13.6 

13.1 

400 

19.8 

19.8 

20.1 

19.7 

19.4 

19.1 

18.6 

18.1 

17.5 

17.0 

16.1 

15.2 

14.8 

420 

19.0 

19.5 

20.0 

20.3 

20.3 

20.3 

20.1 

19.4 

19.0 

18.9 

18.1 

17.3 

16.5 

440 

17.8 

18.7 

19.2 

19.7 

20.1 

20.4 

20.7 

20.7 

20.5 

20.2 

19.8 

19.5 

i8.e 

460 

15.9 

16.8 

17.6 

18.6 

19.2 

19.9 

20.3 

20.6 

21.0 

20.9 

20.9 

20.8 

20.3 

480 

13.5 

14.6 

15.5 

16.6 

17.7 

18.5 

19.3 

19.9 

20.5 

20.8 

21.1 

21.2 

21.2 

500 

11.3 

12.4 

13.4 

14.4 

15.5 

15.5 

17.7 

18.6 

19.1 

19.9 

20.7 

21.0 

21.4 

520 

9.3 

10.2 

11.2 

12.2 

13.3 

14.2 

15.4 

16.4 

17.6 

18.4 

19.2 

19.8 

20.6 

540 

7.9 

8.6 

9.4 

10.1 

11.1 

12.1 

13.1 

14.2 

15.3 

16.3 

17.4 

18.3 

19.2 

500 

6.9 

7.2 

7.8 

8.4 

9.2 

10.1 

11.0 

11.9 

13.1 

14.1 

15.2 

16.2 

17.2 

580 

6.0 

6.3 

6.6 

7.0 

7.6 

8.4 

9.1 

9.9 

10.9 

11.9 

12.9 

14.1 

15.0 

600 

5.6 

5.6 

5.8 

6.1 

6.5 

6.8 

7.4 

8.1 

8.8 

9.9 

10.7 

11.8 

12.8 

620 

5.2 

5.4 

5.3 

5.3 

5.5 

5.9 

6.3 

6.6 

7.2 

8.0 

8.7 

9.5 

10.6 

640 

5.9 

5.6 

5.2 

4.9 

5.0 

5.0 

5.2 

5.5 

5.8 

6.4 

7.0 

7.6 

8.51 

660 

6.9 

6.3 

5.7 

5.4 

5.0 

4.8 

4.5 

4.7, 

4.9 

5.1 

5.5 

6.0 

6.8. 

680 

8.6 

7.6 

6.9 

6.2 

5.6 

5.1 

4.8 

4.6 

4.2 

4.2 

4.5 

4.6 

5.1  i 

700 

11.0 

10.0 

8.7 

7.8 

6.8 

6.3 

5.6 

5.0 

4.6 

4.2 

4.2 

4.0 

4.2 

720 

13.9 

12.5 

11.2 

10.3 

9.1 

7.9 

7.1 

6.2 

5.6 

4.8 

4.5 

4.2 

3.8 

740 

16.8 

15.5 

14.4 

13.0 

11.7 

10.5 

9.4 

8.4 

7.2 

6.5 

5.6 

5.0 

49 

760 

19.7 

18  5 

17.2 

15.9 

14.7  13.5 

12.2 

10.8 

9.8 

8.9 

7.6 

6.7 

&.9 

780 

22.2 

21.2 

20.1 

19.0 

17.6 

16.3 

15.1 

14.0 

12.6 

11.6 

10.2 

9.2 

8.1 

800 

24.4 

23.4 

22.2 

21.3 

20.3 

19.2 

18.0 

16.7 

15.4 

14.3 

13.2 

11.9 

10.8 

820 

25.9 

25.1 

24.4 

23.3 

22.3 

21.6 

20.4 

19.4 

18.2 

17.2 

15.9 

14.6 

13.6 

840 

27.2 

26.6 

25.8 

25.0 

24.3 

23.5 

22.4 

21.6! 

20.5 

19.4 

18.4 

17.3 

16.4 

860 

27.5 

27.1 

26.8 

2G.4 

25.5 

24.8 

24.3 

23.3 

22.2 

21.5 

20.5 

19.6 

18.4 

880 

27.7 

27.5  i 

27.2 

27.0 

20.5 

26.0 

25.5 

24.7  j 

24.1 

23.2 

22.0 

21.4 

20.4 

900 

27.6 

27.8 

27.9 

27.6 

27.1 

26.7 

26.5 

25.7 1 

25.3 

24.6 

23.9 

23.0 

22.0 

920 

27.3 

27.5  1 

27.5 

27.6 

27.7 

27.5 

27.2  1 

26.7 

26.3 

25.7 

25.1 

24.3' 

23.6 

940 

26.1 

26.7! 

27.2 

27.4 

27.7 

27.7 

27.6  i 

27.5  ' 

27.1 

26.6 

26.2 

25.6; 

25.5 

960 

24.7 

25.4 

26.2 

26.6 

27.2 

27.5 

27.7 

27.7 

27.6 

27.4 

27.1 

27.0 

26.2 

980 

22.6 

23.7 

24.6 

25.3 

25.9 

26.8 

27.2 

27.5 

27.7 

27.8  27.6  ^ 

27.5 

27.1 

1000 

20.1 

21.1 

22.2 

23.4 
270 

24.3 

280 

25.2 

25.8 

1 

300  1 

26.6 

310  ! 

27.2 
320 

27.6  27.7 

27.6 

27.6 

240 

250  1 

260 

290 

330  1  340  1 

350 

360 

TABLE  XXX. 


25 


Perturbations  produced  by   Venus. 

Arguments  II.  and  III. 

III. 


1". 

360 

370 

380 

390 

400 

410 

420 

430 

440 

450 

460 
21.3 

1  470 
20.2 

]480 
19.3 

0 

27.6 

27.7 

27.3 

26.7 

26.2 

25.5 

24.7 

23.8 

23.1 

22.3 

20 

27.7 

27.8 

27.8 

27.6 

27.4 

26.8 

26.2 

25.6 

24.8 

24.0 

23.1 

22.0 

20.9 

40 

26.9 

273 

27.6 

27.9 

27.9 

27.7 

27.5 

27.1 

26.3 

25.6 

24.9 

24.0 

23.2 

GO 

25.3 

26.0 

23.8 

27.1 

27.5 

27.9 

27.8 

27.7 

27.3 

27.1 

26.7 

25.9 

25.0 

80 

2.3.1 

24.0 

25.1 

25.9 

26.5 

27.3 

27.5 

27.9 

28.2 

28.0 

27.6 

27.5 

27.2 

100 

20.8 

21.8 

22.6 

23.6 

24.6 

25.5 

26.2 

26.7 

27.2 

27.5 

27.6 

27.8 

27.4 

120 

18.5 

19.6 

20.6 

21.5 

22.4 

23.2 

24.1 

25.1 

25.8 

26.4 

26.9 

27.3 

27.5 

140 

17.1 

17.9 

IS. 6 

19.3 

20.3 

21.3 

22.0 

22.9 

23.7 

24.7 

25.5 

26.0 

26.7 

IGO 

16.5 

17.1 

17.4 

18.1 

18.8 

19.3 

20.1 

21.0 

21.9 

22.6 

23.5 

24.2 

25.1 

ISO 

16.9 

17.0 

17.1 

17.4 

18.0 

IS. 4 

18.9 

19.4 

20.1 

20.7 

21.2 

22.2 

23.0 

200 

17.7 

17.5 

17.7 

17.7 

17.6 

18.1 

18.3 

18.7 

19.2 

19.7 

20.1 

20.8 

21.5 

220 

18.3 

18.2 

18.3 

18.3 

18.3 

18.3 

18.6 

18.7 

18.9 

19.3 

19.5 

20.0 

20.4 

240 

18.6 

18.8 

18.9 

18.9 

18.9 

19.0 

19.2 

19.1 

19.2 

19.5 

19.6 

19.7 

19.9 

260 

18.2 

18.5 

18.7 

18.8 

19.0 

19.3 

19.5 

19.6 

19.9 

19.9 

20.0 

20.1 

20.2 

280 

16.8 

17.4 

17.9 

18.3 

18.7 

19.1 

19.3 

19.8 

20.0 

20.2 

20.4 

20.6 

20.8 

300 

15.5 

15.8 

16.2 

16.6 

17.6 

18.1 

18.5 

19.2 

19.4 

19.9 

20.6 

20.8 

20.9 

320 

13.8 

14.2 

14.6 

15.1 

15.6 

16.2 

16.8 

17.7 

18.3 

18.9 

19.5 

20.1 

20.8 

340 

12.6 

12.9 

13.0 

13.3 

13.7 

14.4 

14.9 

15.5 

16.2 

17.1 

18.0 

18.6 

19.4 

360 

12.3 

12.1 

11.9 

12.0 

12.3 

12.5 

13.0 

13.4 

14.2 

14.9 

15.7 

16.5 

17.3 

3S0 

13.1 

12.5 

11.9 

11.6 

11.5 

11.4 

11.6 

11.7 

12.3 

12.7 

1.3.3 

14.0 

15.0 

400 

14.8 

13.9 

13.1 

12.5 

11.7 

11.2 

11.1 

10.9 

11.0 

11.1 

11.4 

12.0 

12.6 

420 

16.5 

15.7 

15.1 

14.3 

13.4 

12.5 

11.7 

11.1 

10.8 

10.8 

10.5 

10.6 

10.7 

440 

18.6 

17.9 

17.1 

16.1 

15.6 

14.4 

13.5 

12.8 

11.9 

11.1 

10.6 

10.3 

10.3 

400 

20.3 

19.8 

19.3 

18.5 

17.6 

16.8 

15.9 

14.7 

13.7 

12.9 

12.0 

11.1 

10.9 

430 

21.2 

21.1 

20.8 

20.3 

19.7 

19.1 

18.3 

17.4 

10.4 

15.0 

14.1 

13.2 

12.2 

500 

21.4 

21.4 

21.4 

21.3 

21.1 

20.8 

20.0 

19.5 

18.8 

17.8 

17.0 

15.7 

14.4 

520 

20.6 

21.2 

21.7 

21.7 

21.5 

21.5 

21.4 

21.1 

20.5 

19.8 

19.1 

18.2 

17.6 

540 

19.2 

20.0 

20.7 

21.1 

21.8 

22.0 

21.8 

21.7 

21.5 

21.2 

20.9 

20.3 

19.6 

500 

17.2 

18.4 

19.0 

20.0 

20.8 

21.1 

22.7 

21.9 

22.2 

22.1 

21.9 

21.7 

21.1 

530 

15.0 

16.0 

17.3 

18.2 

19.1 

19.9 

20.8 

21.1 

21.7 

22.0 

22.2 

22.3 

22.1 

GOO 

12.8 

13.9 

15.1 

15.9 

17.2 

18.0 

19.0 

19.9 

20.6 

21.3 

21.8 

22.0 

22.4 

620 

10.6 

11.5 

12.7 

13.7 

14.9 

16.0 

17.1 

18.3 

19.1 

19.9 

20.8 

21.3 

22.0 

640 

8.5 

9.5 

10.4 

11.3 

12.3 

13.7 

14.9 

16.0 

17.1 

18.1 

19.0 

19.9 

20.7 

660 

6.8 

7.4 

8.2 

9.1 

10.1 

11.1 

12.2 

13.6 

14.6 

15.8 

17.1 

18.1 

19.0 

6S0 

5.1 

5.7 

6.4 

7.1 

7.9 

8.7 

9.7 

11.0 

12.1 

13.1 

14.1 

15.7 

10.8 

700 

4.2 

4.4 

4.7 

5.1 

5.8 

6.7 

7.4 

8.4 

9.4 

10.6 

11.5 

13.0 

14.1 

720 

3.8 

3.8 

3.8 

4.0 

4.4 

4.8 

5.4 

5.9 

6.9 

8.0 

9.1 

10.1 

11.5 

740 

4.3 

3.9 

3.8 

3.7 

3.6 

3.8 

3.9 

4.4 

4.9 

5.7 

6.4 

7.4 

8.9 

750 

5.9 

5.1 

4.4 

4.0 

3.6 

3.4 

3.4 

3.5 

3.9 

4.3 

4.7 

5.2 

5.9 

780 

8.1 

7.1 

6.1 

5.3 

4.6 

4.1 

3.7 

3.3 

3.3 

3.1 

3.4 

3.6 

4.1 

800 

10.8 

9.7 

8.5 

7.5 

6.5 

5.6 

4.9 

4.2 

3.8 

3.4 

3.2 

3.1 

3.1 

820 

13.6 

12.5 

11.2 

10.1 

9.0 

8.0 

6.9 

6.1 

5.3 

4.7 

3.9 

3.7 

3.1 

840 

16.4 

15.1 

13.7 

12.9 

11.7 

10.6 

9.5 

8.6 

7.5 

6.6 

5.7 

4.9 

4.4 

860 

18.4 

17.5 

16.6 

15.4 

14.3 

13.1 

12.1 

11.1 

10.0 

9.1 

7.9 

7.0 

6.3 

880 

20.4 

19.6 

18.7 

17.5 

16.6 

15.6 

14.5 

13.6 

12.5 

11,5 

10.4 

9.5 

8.6 

900 

22.0 

21.1 

20.2 

19.4 

18.7 

17.7 

16.5 

15.7 

14.7  j 

13.8 

12.5 

11.9 

10.9 

920 

23.6 

22.7 

21.7 

21.1 

20.1 

19.4 

18.4 

17.5 

16.7  1 

15.6 

14.8 

13.9 

13.1 

940 

25.5 

24.1 

23.4 

22.4 

21.4 

20.6 

19.9 

19.0 

18.2 

17.3 

16.6 

15.71 

14.8 

960 

26.2 

25.6 

24.7 

24.1 

23.3 

22.3 

21.3 

20.6  j 

19.0  18.9 

17.9 

17.1  j 

16.3 

980 

27.1 

26.7 

26.3 

25.5 

24.9 

23.8 

23.4 

22.2' 

21.0  20.4 

19.4 

18.6  1 

17.7 

1000 

27.6 

27.7 

27.3 

26.7 

26.2 

25.5 

24.7 

23.8, 

23.1  22.3 

21.3 

20.2 

470  1 

19.3 

480  j 

360 

370 

380 

390 

400  i  410  1 

420  1 

430 

440 

450  ' 

460 

TABLE  XXX. 


Perturbations  produced  by  Venus. 

Arguments  II  and  III. 

III. 


II. 

480 
19.3 

490 
IS. 3 

500 

510 

,  520 

530 

i  540 

550  1  560  j  570 

1  580 
11.7 

|690 

coo' 

0 

17.4 

16.6 

15.7 

15.0  14.2 

13.6 

13.1 

12.3 

|11.3  10..? 

20 

20.9 

202 

19.1 

18.2 

17.1 

16.2  15.5 

14.7 

14.1 

13.3 

12.7 

122  11.5 

40 

23.2 

22.0 

20.8 

20.1 

18.9 

17.9  17.1 

15.9 

15.1 

14.4 

13.7 

130  12.G 

60 

25.0 

I  24.0 

23.2 

22.0 

20.7 

19.9  18.9 

17.7 

16.8 

15.8 

14.9 

140  13.2 

80 

27.2 

1  2G.4 

25.6 

24.1 

23.2 

22.1  20.8 

20.0 

18.7 

17.9 

16.6 

15.6  14.^ 

100 

27.4 

27.2 

26.8 

26.3 

25.4 

24.5  23.5 

22.2 

20.9 

20.0 

18.6 

17.6 

j  IG.C 

120 

27.5 

27.5 

27.6 

27.1 

20.8 

26.3  25.4 

24.6 

23.7 

22.4 

21.0 

20.1 

IS.- 

140 

26.7 

27.0 

27.2  27.4 

27.3 

27.4  20.9 

26  2 

25.4 

24.6 

239 

22.6  2.11 

160 

25.1 

25.0 

26.1 

26.7 

26.9 

27.3  27.1 

27.0 

26.9 

26.4 

25.5 

24.7 

23.9 

180 

23.0 

23.8 

24.5 

25.0 

25.7 

26.3  26.7 

26.8 

27.0 

26.8 

26.6 

26.2 

25.6 

200 

21.5 

22.2 

22.8 

123.5 

24.1 

24.7,25.5 

25.8 

26.3 

26.6 

26.6 

26.6 

26.4 

220 

20.4 

21.0 

21.5 

22.0 

22.6 

23.2  23.8 

24.5 

25.0 

25.4 

25.8 

26.0 

26.2 

240 

19.9 

20.4 

20.8 

21.2 

21.6 

21.8  22.2 

22.6 

23  1 

23.3 

23.9 

24.2 

24.6 

260 

20.2 

20.3 

20.6 

21.2 

21.4 

21.7:21.9 

22.2 

22  3 

22.7 

23.1 

23.3 

23.6 

280 

20.8 

20.8 

21.0 

21.1 

21.3 

21.4  21.5 

21.8 

22.0 

22.2 

22.7 

230 

23.3 

300 

20.9 

21.0 

21.5 

21.7 

21.7 

22.0  22.0 

1 

22.1 

22  1 

00  0 

22  4 

22.6 

22.8 

320 

20.8 

21.2 

21.5 

21.6 

22.0 

22.3 

22.5 

22.5 

22  6 

22.7 

22.8 

22.8 

22.9 

340 

19.4 

20.2 

20.8 

21.5 

21.9 

22.1 

'22.6 

23.0 

23.2 

23.4 

23.3 

23.4 

23.5 

360 

17.3 

18.4 

19.5 

20  0 

20.6 

21.5 

22.2 

22.7 

23.0 

23.7 

23.7 

24.0 

24.2 

380 

15.0 

15.9 

16.9 

17.8 

18.6 

19.6 

20.6 

21.5 

22.3 

22.9 

23  5 

23.9 

24.5 

400 

12.6 

13.2 

14.2 

15.4 

16.2 

17.3 

18.1 

19.2 

20.3 

21.4 

22  4 

23.0 

23.7 

420 

10.7 

11.2 

12.0 

12.5 

13.5 

[4.5 

15.6 

16.7 

17.7 

18.7 

20  1 

21.0 

22.0 

440 

10.3 

10.2 

10.3 

10.5 

11.3 

12.0 

12.9 

13.6 

147 

16.0 

17.0 

IS. 3 

19.5 

460 

10.9 

10.1 

9.9 

9.9 

9.9 

10.1 

10.7 

113 

12.2 

130 

140 

1.5.1 

16.5 

480 

12.2 

11.4 

10.7 

10.1 

9.7 

9.5 

9.7 

9.9 

10.2 

10.7 

11.7 

12.5 

13  4 

500 

14.4 

13.6 

12.5 

11.6 

10.9 

10.2 

9.8 

9.4 

9.3 

9.6 

9.8 

10.2 

11.1 

520 

17.6 

16.2 

15.1 

13.9 

12.9 

11.9 

10.9 

10.3 

9.8 

9.5 

9.2 

9.2 

9.6 

540 

19.6 

18.6 

18.0 

16.7 

15.4 

145 

12.2 

123 

11.3 

105 

10.1 

95 

9.3 

560 

21.1 

20.4 

19.8 

19.0 

18.2 

17.2 

16.0 

14,8 

13.7 

12.7 

11.7 

10.9 

102 

580 

22.1 

21.8 

21.5 

20.9 

20.3 

19.3 

18.6 

17.3 

16.5 

15.4 

14.0 

J29 

12.2 

600 

22.4 

22.4 

22.2 

22.2 

21.5 

21.2 

20  6 

19.5 

19.1 

17.7 

16.8 

15.8 

14.4 

620 

22.0 

22.3 

22.4 

22.4 

22.3 

22.3 

21  9 

21.5 

20.9 

20.0 

19.3 

18.0 

16.9 

640 

20.7 

21.7 

22.0 

22.3 

22.6 

22.5 

22  6 

22.4 

22.0 

21.6 

21.1 

20  3 

196 

660 

19.0 

20.0 

20.8 

21.3 

22.1 

22.3 

22  6 

22.8 

22.7 

226 

22.2 

21.8 

21.3 

680 

16.8 

18.0 

19.0 

19.9 

20.8 

21.5 

22  1 

22.6 

22.7 

23.0 

23.0 

22.8 

22.4 

700 

14.1 

15.2 

16.8 

17.9 

18.8 

20.0 

22  1 

21.5 

22  2 

22.6 

22.9 

23  0 

23.2 

720 

11.5 

12.7 

13,9 

15.0 

16.4 

17.9 

18.6 

19.7 

20.8 

21.6 

22  3 

Of>  y 

23.0 

740 

8.9 

9.8 

10.9 

12.2 

13.6 

14.8 

16.2 

17.5 

18.7 

19.5 

20.6 

21.6 

22.3 

760 

5.9 

6.8 

8.0 

9.3 

103 

11.8 

132 

14.5 

15.9 

17.4 

18  2 

19.5 

205 

780 

4.1 

4.9 

5.6 

6.4 

7.5 

8.6 

9.9 

11.1 

12.6 

14.0 

15.6 

16.8 

18.1 

800 

3.1 

3.3 

4.4 

4.8 

5.5 

6.1 

6.9 

7.9 

9.4 

10.7 

12.1 

13.4 

14.9 

820 

3.1 

3.1 

3.2 

3.1 

3.6 

3.9 

4.8 

5.7 

65 

75 

8.7 

10.0 

11.5 

840 

4.4 

3.7 

3.5 

3.2 

3.2 

3.1 

3.4 

3.7 

4.1 

5.0 

6.2 

7.0 

8.3 

860 

6.3 

5.5 

4.6 

4.1 

3.6 

3.4 

3.3 

3.2 

3.4 

3.4 

4.0 

4.5 

0.6 

880 

8.6 

7.6 

6.7 

5.9 

5.2 

4.5 

4.1 

3.8 

3.5 

3  4 

34 

3.6 

3.9 

900 

10.9 

10.0 

9.1 

6.3 

7.2 

6.5 

5.8 

5.1 

4.4 

42 

3.8 

3.6 

3.6 

920 

13.1 

12.1 

11.2 

10.3 

9.6 

8.7 

7.7 

6.9 

6.3 

5.8 

5.1 

4.6 

4  2 

940 

14.8 

14.1 

13.1 

12.4 

11.5 

10.8 

9.8 

9.1 

8.3 

7.6 

6.8 

6.5 

5.9 

960 

16.3 

15.4 

14.6 

14.0 

13.2 

12.6  11.7 

11.0 

10.1 

96 

88 

8.1 

75 

980 

17.7 

16.8 

16.2 

15.2 

14.5 

13.9  13.1 

12  5 

11.8 

11.2 

105 

9.7 

93 

1000 

19.3 
480 

18.3 

17.4 

16.6 

15.7 

15.0  14.2 

13.6 

13.1 

12.3  11.7 

1 

11.3 

10.8 

1 

490 

500 

510 

520 

530 

540 

5.50 

500 

570 

580 

590  1 

COO 

TABLE  XXX. 


27 


Perivrhaticns  produced  hy   Venus. 

Arguments  II.  and  III. 

III. 


II.   600  610  620  630 

1  640  ]  650 

660 

670  680  690 

700 

710 

720 

0  lO.S 

10.2 

9.5  9.1 

8.4 

7.9 

7.4 

7.0  6.6 

;  6.3 

5.9 

5.5 

5.4 

20  11.5 

11.3 

10.7-10.4 

9.8 

9.4 

8.9 

8.5  7.9 

7.7 

7.3 

6.7 

6.6 

40  10.3 

12.0 

111.5 

11.0 

10.7 

10.3 

10.0 

9.6   9.3   8.9 

8.5 

8.1 

7.8 

60  l:J.:3 

12.7 

12.1 

11.6 

11.2 

10.9 

10.5 

10.2  10.0 

i  9.8 

9.5 

9.2 

8.9 

80  M.3 

13.6 

12.9 

12.4 

11.8 

11.3 

10.9 

10.7  10.3 

!  9.9 

9.8 

9.8 

9.6 

100 

IG.G 

15.4 

14.4 

13.4 

12.6 

12.1 

11.5 

11.0  10.6 

,  10.2 

10.0 

9.9 

9.6 

120 

IS.S 

17.7 

16.4 

15.3 

14.3 

13.2 

12.4 

11.6  11.2 

10.6 

10.1 

10.1 

9.6 

14G 

21.1 

20.1 

18.9 

17.7 

16.5 

15.2 

14.2 

13.0  12.3 

11.6 

11.1 

10.3 

9,9 

160 

23.0 

22.9 

21.5 

20.4 

19.2 

17.9 

16.6 

15.3  14.1 

13.1 

12.0 

11.2 

105 

ISO 

25.G 

21.8 

23.9 

22.9 

21.6 

20.6 

19.1 

18.0  16.7 

15.5 

14.3 

12.9 

12,0 

200 

2G.4 

26.0 

25.6 

24.9 

24.0 

22.9 

21.7 

20.8^  19.3 

18.1 

1G.9 

15.5 

14.4 

220 

2G.2 

26.3 

26.1 

25.8 

25.3 

24.9 

24.1 

23.1 

21.2 

20.9 

19.7 

18.3 

17.1 

210 

24.  G 

25.1 

25.1 

25.3 

25.2 

25.1 

24.7 

24.3  24.0 

23.0 

21.9 

21.3 

20.2 

260 

23.6 

23.9 

24.2 

24.5 

24.7 

24.8 

24.9 

24.6  24.3 

23.8 

23.4 

22.9 

21.6 

2S0 

23.3 

23.6 

23.9 

24.2 

24.7 

24.8 

25.0 

24.9  \  24.9 

24.8 

24.4 

24.0 

23.5 

300 

22.8 

23.0 

23.3 

23.4 

23.8 

24.0 

24.1 

24.5 

24.5 

24.6 

24.5 

24.4 

24.0 

320 

22.9 

230 

23.1 

23.2 

23.4 

23.3 

23.6 

23.8 

24.0 

23.9 

24.2 

24.2 

24.2 

340 

23.5 

23.. 5 

235 

23.4 

23.5 

23.6 

23  6 

23.5 

23.5 

23.6 

23.9 

23.8 

23.8 

360 

24  2 

24  2 

24.3 

24.2 

24.2 

24.0 

23.7 

23.9 

24.0 

23.7 

23.7 

23.6 

23.6 

3S0 

24  5 

24.6 

24.8 

25.1 

24.8 

24.9 

25.0 

24.9 

24.6 

24.5 

24.5 

24.3 

24.0 

400 

23.7 

24.3 

24.7 

25.0 

25.4 

25.7 

25.7 

25.5 

25.5 

25.4 

25.2 

24.8 

24.6 

420 

22.0 

23.0 

23  7  i  24.6 

25.0 

25.7 

26.1 

26.2 

26.3 

26.5 

26.2 

26.0 

25.9 

440 

19.5 

20  S 

21.7 

22.7 

23.7 

24  6 

25.4 

26.0 

26.5 

26.7 

26.9 

27.0 

26.9 

460 

1G.5 

17.8 

100 

20.1 

21.4 

223 

23.5 

24.8 

25.4 

26.1 

26.7 

27.1 

27.3 

4S0 

13.4 

14.5 

15.6 

17.0 

18.5 

19.7 

20.9 

22.1 

23.2 

24.4 

25.4 

20.2 

26.8 

500 

11.1 

120 

13.0 

13.8 

14.9 

16.3 

17.9 

19.1 

20.5 

21.6 

22.9 

24.2 

25.1 

520 

96 

98 

10.5 

11.5 

12.4 

13.4 

14.4 

15.5 

17.1 

18.4 

19.9 

21.2 

22.3 

540 

9.3 

9.0 

9.2 

9.6 

10.3 

11.0 

11.9 

12.8 

13.9 

15.1 

16.5 

17.9 

19.4 

560 

10  2 

9.7 

9.3 

9.1 

9.1 

9.4 

10.0 

10.6 

11.5 

12.4 

13.3 

14.5 

16.0 

530 

12.2 

11.3 

10.4 

9.9 

9.4 

9.0 

9.2 

9.3 

9.7 

10.4 

11.0 

12.0 

12.7 

GOO 

14.4 

13.3 

12.5 

11.6 

10.8 

10.1 

9.6 

9.4 

9.1 

9.3 

9.9 

10.0 

10.8 

G20 

16. 9 

16.1 

14.9 

13.7 

12.7 

12.0 

11.1 

10.4 

9.8 

9.5 

9.5 

9.3 

9.7 

610 

19  G 

18.4 

17.4 

163 

15.2 

14.2 

13.1 

12.1 

11.3 

10.6 

10.1 

9.6 

9.5 

660 

21.3 

206 

19.9 

18.7 

17.8 

16.7 

15.6 

14.4 

134 

12.4 

11.7 

11.0 

10.2 

6^0 

22  4 

22.0 

21.5 

208 

20.2 

19.0 

18.1 

17.0 

15.8 

14.7 

13.7 

12.8 

12.0 

700 

23.2 

23.2 

22.6 

22.2 

21.7 

21.0 

20.5 

19.3 

18.3 

17.3 

16.0 

15.0 

14.1 

720 

23.0 

233 

23.2 

23.4 

23.1 

22.4 

21.9 

21.3 

20.8  1 

19.5 

18.5 

17.6 

16.4 

740 

22.3 

22  8 

23,2 

23.4 

23.6 

23.6 

23.3 

22.8 

22  2 

21.6 

21.1 

19.9 

18.8 

760 

20.5 

21.4 

225 

22.8 

233 

23.7 

236 

23.8 

23.5  23.3 

22.7 

21.8 

21.3 

780 

18  1 

192 

20.4 

21.3 

22.3 

23.0 

23.3 

23.7 

23.8  24,0 

23.8 

23.5 

23.0 

800 

14.9 

16.4 

17.7  J 

19.1 

20.1 

21.2 

21.1 

22.9 

23.4  23.8 

24.1 

24.2 

23.9 

820 

11.5 

12.9 

14.3 

15.8 

17.8 

18.7 

20.0 

20.9 

22.0  22.7 

23.5 

23.9 

24.0 

840 

8.2 

9.5 

10.8 

12.2 

13.8 

15.2 

16.6 

18.1 

19.5  20.G 

21.7 

22.6 

23.3 

sno 

5.6 

6.8 

7.7 

8.8 

10.2 

11.5 

13.2 

14.7 

16.0  17.4 

19.0 

20.2 

21.3 

880 

3.9 

4.4 

5.2 

6.1 

7.2 

8.2 

9.7 

10.9 

12.5  14.1 

15.4 

16.8 

18.2 

900 

3.6 

3.6 

3.9 

4.2 

5.0 

5.7 

6.6 

7.8 

9.1 

10.3 

11.8 

13.4 

14.8 

920 

4.2 

3.8 

3.9 

3.9 

4.0 

4.3 

4.7 

5.4 

6.4 

7.3 

8.6 

9.8 

11.2 

940 

5.9 

5.1 

4.6 

44 

4.2 

4.3 

43 

4.3 

4.9 

5.3 

6.3 

7.0 

8.0 

9G0 

7.5 

6.9 

6.3 

58 

53 

4.7 

4.7 

4.6 

4.6 

4.6 

4.9 

5.4 

6.0 

930 

9.3 

8.7 

7.9 

7.4 

6.8 

6.4 

6.0 

5.6 

5.2 

5.0 

4.9 

5.1 

5.1 

1000 

10  8 

600  ! 

10.2 

9.5 

9.1 

8.4 

7.9 

7.4 

7.0 

6.6 

6.3 

5.9 

5.5 

710 

5.4 

720 

GIO  020 

C30 

640  ! 

650  \   660  \  G70  1 

680 

690  ' 

700  1 

28 


TABLE  XXX. 


Perturbations  produced  hy   Venus. 
Arguments  II.  and  III. 

m. 


11. 

0 

720 

730 

740 

750 

760 

770 
6.8 

780 

790 

800 

810 

820 

830 

840 

5.4 

5.5 

5.8 

6.0 

6.3 

7.6 

8.4 

9.3 

10.4 

11.7 

12.9 

14.3 

20 

6.6 

6.3 

6.0 

6.1 

6.1 

6.2 

6.5 

6.9 

7.7 

8.3 

9.4 

10.2 

11.2 

40 

7.8 

7.4 

7.1 

7.0 

6.7 

6.6 

6.8 

6.8 

6.9 

7.2 

7.7 

8.5 

9.3 

60 

8.9 

8.8 

8.3 

8.1 

7.8 

7.6 

7.4 

7.4 

7.3 

7.4 

7.4 

7.7 

8.3 

80 

9.6 

9.5 

9.1 

9.1 

9.0 

8.8 

8.4 

8.2 

8.1 

8.1 

8.0 

8.1 

8.2 

100 

9.6 

9.5 

9.6 

9.5 

9.5 

9.3 

9.3 

9.2 

9.2 

9.0 

8.7 

8.7 

8.7 

120 

9.6 

9.6 

9.5 

9.3 

9.4 

9.6 

9.6 

9.5 

9.5 

9.6 

9.6 

9.6 

9.6 

140 

9.9 

9.5 

9.6 

9.4 

9.3 

9.3 

9.0 

9.3 

9.5 

9.8 

9.7 

9.8 

10.0 

160 

10.5 

9.9 

9.5 

9.1 

8.9 

9.0 

8.9 

9.0 

9.0 

9.0 

9.5 

9.6 

9.9 

ISO 

12.0 

11.0 

10.1 

9.7 

9.1 

8.8 

8.7 

8.3 

8.5 

8.7 

8.S 

9.0 

9.1 

200 

14.4 

13.3 

12.0 

11.0 

10.1 

9.4 

8.9 

8.5 

8.2 

8.0 

8.0 

8.3 

8.5 

220 

17.1 

15.7 

14.6 

13.2 

12.0 

10.9 

10.2 

9.2 

8.7 

8.3 

7.9 

7.7 

7.7 

240  20.2 

19.1 

17.8 

16.5 

14.5 

13.4 

12  2 

11.1 

10.0 

9.4 

8.4 

8.0 

7.7 

260  21.6 

21.1 

20.1 

19.2 

17.3 

15.9 

14.6 

13.4 

12.4 

11.3 

10.1 

9.1 

8.6 

280  23.5 

22.7 

21.6 

21.0 

19.8 

18.8 

17.3 

16.1 

15.0 

13.5 

12.5 

11.5 

10.2 

300  24.0 

23.4 

23.2 

22.4 

21.4 

20.5 

19.8 

18.7 

17.5 

16.1 

15.0 

13.7 

12.4 

320  24.2 

23.9 

23.5 

23.1 

22.7 

22.2 

21.2 

20.6 

19.6 

18.6 

17.5 

16.3 

15.1 

340  23.8 

23.9 

23.7 

23.5 

23.2 

22.8 

22.3 

21.4 

20.9 

20.5 

19.2 

18.6 

17.4 

360 

23.6 

23.6 

23.6 

23.3 

23.3 

23.1 

22.9 

22.4 

22.0 

21.4 

20.4 

19.9 

18.9 

380 

24.0 

24.0  23.7 

235 

23.3 

23.1 

23.1 

22.7 

22.4 

22.2 

21.6 

20.8 

20.0 

400 

24.6 

24.4  '  24.4 

24.0 

23.8 

23.4 

23.2 

23.0 

22.8 

22.4 

22.1 

21.6 

21.3 

420 

25.9 

25.6  25.2 

24.8 

24.7 

24.3 

23.9 

23.6 

23.3 

22.9 

22.7 

22.3 

21.7 

440 

26.9 

26.6  26.4 

23.2 

25.9 

25.5 

25.2 

24.9 

24.5 

23  8 

23.4 

23.0 

22.8 

460 

27.3 

27.6  27.6 

27.4 

27.0 

26.9 

26.5 

28.1 

25.6 

25.0 

24.6 

24.2 

23.7 

480 

26.8 

27.4 

27.6 

28.0 

28.1 

28.2 

27.7 

27.4 

27.3 

26.6 

26.2 

25.7 

25.1 

500 

25.1 

26.1 

26.8 

27.5 

28.1 

28.2 

28.6 

28.5 

28.4 

28.3 

27.6 

27.2 

26.7 

520 

22.3 

23.9  '  24.8 

25.9 

26.8 

27.5 

28.1 

28.5 

28.7 

29.0 

28.8 

28.6 

28.4 

540 

19.4 

20.7  22.1 

23.4 

24.6 

25.6 

26.5 

27.4 

28.0 

28.7 

28.9 

29.1 

29.2 

560 

16.0 

17.3  18.6 

19.9 

21.4 

22.9 

24.1 

25.5 

26.4 

27.3 

28.2 

28.6 

29.2 

580 

12.7 

14.1  15.5 

16.8 

18.0 

19.3 

20.9 

22.2 

23.5 

24.9 

26.1 

27.0 

27.8 

600 

10.8 

11.6  12.7 

13.6 

14.9 

16.2 

17.5 

18.7 

20.2 

21.8 

23.0 

24.4 

25.5 

620 

9.7 

10.0  10.5 

10.7 

12.2 

13.2 

14.4 

15.6 

17.0 

18.3 

19.6 

21.2 

22.6 

640 

9.5 

9.4   9.6 

10.1 

10.4 

11.1 

12.0 

13.0 

14.0 

15.2 

16.5 

17.9 

19  2 

660 

10.2 

10.0   9.7 

9.5 

9.5 

9.9 

10.4 

11.0 

11.7 

12.7 

13.8 

14.9 

16.2 

680 

120 

11.2  10.5 

10.0 

9.7 

9.5 

9.6 

10.0 

10.4 

11.0 

11.6 

12.5 

13.8 

700 

14.1 

13.1  13.3 

11.3 

10.7 

10.1 

9.7 

9.7 

9.9 

9.9 

10.4 

10.9 

11.5 

720 

16.4 

15.3 

14.4 

13.3 

12.2 

11.6 

10.9 

10.2 

10.1 

9.9 

10.0 

10.1 

10.4 

740 

18.8 

17.7 

16.7 

15.6 

14.4 

13.5 

12.4 

11.0 

11.1 

10.7 

10.1 

10.0 

10.3 

760 

21.3 

20.1 

19.2 

18.1 

16.6 

15.6 

14.7 

13.6 

12.8 

11.9 

11.3 

10.7 

10.3 

780 

23.0 

22.3 

21.5 

20.5 

19.4 

18.4 

17.2 

iri.8 

14.9 

14.0 

13  0 

12.2 

11.3 

800 

23.9 

23.9 

23.4 

22.6 

21.9 

20.7 

19.8 

IS.  3 

17.5 

16.3 

15.1 

14.2 

13.4 

820 

24.0 

24.5 

24.2 

23.9 

23.3 

22.6 

22.3 

21.3 

20.3 

19.4 

18.3 

17.3 

16.2 

840 

23  3 

24.0 

24  3 

24.5 

24.4 

24.3 

23.8 

23.4 

22.7 

21.7 

20.8 

19.6 

18.3 

860 

21.3 

22.3 

23.3 

23.9 

24.2 

24.7 

24.5 

24.5 

24.3 

23.6 

23.1 

21.9 

21.0 

880 

182 

19.7 

20.9 

22.0 

22.8 

23.8 

24  i 

24.6 

24.8 

24.7 

34.5 

24.0 

23.5 

900 

14.8 

16.1 

17.6 

19.0 

20.6 

21.5 

22.5 

23.2 

24.1 

24.5 

24.2 

24.8 

24.5 

920 

11.2 

12.6 

14.0 

15.5 

17.0 

18.4 

19.9 

21.0 

22.0 

22.9 

235 

24.5 

24.5 

940 

8.0 

9.3 

10.7 

12.0 

13.3 

14.8 

16.4 

17.6 

19.1 

20.4 

21.4 

22.4 

23.2 

960 

6.0 

6.9 

7.8 

8.6 

10.2 

11.5 

12.7 

14.1 

15.6 

16.9 

18.5 

19.5 

20.7 

980 

5.1 

5.5 

6.0 

6.7 

7.7 

8.5 

9.7 

10.9 

12.2 

13.6 

14.8 

16.1 

17.6 

1000 

5.4 

720 

5.5  5.8 

5.8 
750 

6.3 

6.8 

7.6 

8.4 
790 

9.3 

10.5 
810 

11.7 
820 

12.9 

14.3 

730 

740 

760 

770 

780 

800 

830 

840 

TABLE  XXX. 


29 


Perturbations  'produced  by   Venus. 

Arguments  II.  and  III. 

III. 


11. 

0 

810 

850 
15.5 

860 

870 

880 

890 
20.2 

900 

910 

920 

930 
23.5 

940 

950 
24.2 

960 
24.2 

14.3 

16.9 

18.2 

19.2 

21.4 

22.5 

23.0 

24.0 

20 

11.2 

12.4 

13.6 

14.9 

18.2 

17.3 

18.6 

19.6 

20.5 

21.5 

22.4 

23.1 

23.6 

40 

9.3 

10.2 

10.9 

11.8 

13.3 

14.2 

15.5 

16.6 

17.8 

18.8 

19.7 

20.7 

21.6 

60 

8.3 

8.7 

9.5 

10.1 

10.8 

11.6 

12.7 

13.8 

14.9 

15.9 

17.0 

18,1 

19.1 

80 

8.2 

S.3 

8.6 

8.9 

9.6 

10.3 

10.7 

11.6 

12.5 

13.3 

14.5 

15,2 

16.2 

100 

8.7 

8.7 

8.9 

9.0 

9.1 

9.4 

9.9 

10.4 

11.0 

11.7 

12.4 

12.9 

14.0 

120 

9.6 

9.5 

9.3 

9.6 

9.6 

9.7 

9.9 

9.8 

10.4 

10.9 

11.3 

11.8 

12.3 

40 

10.0 

10.2 

10.1 

10.2 

10.1 

10.3 

10.4 

10.5 

10.5 

10.6 

10.9 

11.4 

11.5 

100 

9.9 

10.0 

10.2 

10.4 

10.6 

11.0 

11.0 

10.9 

11.0 

11.3 

11.3 

11.3 

11.6 

180 

9.1 

9.6 

99 

10.1 

10.4 

10.7 

11.0 

11.3 

11  5 

11.7 

11.7 

11.9 

12,2 

200 

8.5 

8.8 

9.1 

9.5 

9.7 

10.0 

10.5 

11.0 

11.2 

11.6 

12.0 

12.2 

12,4 

220 

7.7 

7.7 

8.1 

8.4 

8.8 

9.2 

9.7 

10.1 

10.6 

11.0 

11,4 

11.8 

12,3 

240 

7.7 

7.3 

7.4 

7.4 

7.7 

8.0 

8.4 

9.0 

9.6 

100 

10.5 

11.0 

11.5 

2G0 

8.6 

7.9 

7.4 

7.2 

7.1 

7.1 

7.3 

7.6 

8.1 

8.5 

9.3 

10.0 

10.4 

280 

10.2 

9.2 

8.3 

7.9 

7.4 

7.1 

7.0 

6.9 

7.0 

7.3 

7.7 

8.5 

8.8 

300 

12.4 

11.4 

10.4 

9.3 

8.5 

7.8 

7.4 

6.9 

6.7 

6.8 

6.8 

7.0 

7.5 

320 

15.1 

13.9 

12.5 

11.4 

10.5 

9.7 

8.6 

7.8 

7.4 

7.0 

6.6 

05 

6.7 

340 

17.4 

16.4 

15.2 

13.9 

12.7 

11.6 

10.6 

9.7 

8.7 

8.0 

7.3 

6.8 

6.6 

.SCO 

18.9 

18.1 

17.4 

16.3 

15.1 

13.8 

12  8 

11.7 

10.6 

9.8 

8.8 

8.0 

7.4 

380 

20.0 

196 

18.8 

17.7 

16.9 

13.0 

15.1 

13.9 

12.7 

11.8 

10.8 

9.8 

8.9 

400 

21.3 

20.6 

19.6 

19.4 

18.4 

17.6 

16.5 

15.7 

14.8 

13.7 

12.8 

11.8 

10.9 

420 

21.7 

21.1 

20.8 

20.3 

19.3 

18.9 

18.2 

17.2 

16.3 

15.3 

14.5 

13.7 

12.6 

440 

22.8 

22.1 

21.6 

20  8 

20.6 

19.7 

19.0 

18.6 

17.7 

16  6 

15.9 

15.1 

14.2 

460 

23.7 

23  3 

22.7 

22.0 

21.6 

20.9 

20.2 

19.5 

18.5 

18.1 

17.3 

16.7 

15.7 

480 

25.1 

24.4 

239 

23.3 

22.8 

22.0 

21.4 

20.9 

20  2 

19.3 

18.3 

17.7 

169 

500 

26.7 

26.3 

25.7 

24.9 

24.3 

23.6 

23.0 

22.3 

21.4 

20.7 

20.3 

19.1 

18.1 

.520 

28.4 

27.8 

27  3 

26.8 

26.3 

25.6 

24.7 

23.9 

23.3 

22.6 

21.8 

20,8 

20.1 

.'540 

29.2 

29.2 

28.9 

28.5 

27.8 

27.4 

26,8 

26.1 

25.3 

24.4 

23.7 

23,0 

220 

560 

29.2 

29  3 

29.5 

29.6 

29.3 

29.1 

28.8 

28.0 

27.4 

26.9 

26.1 

25,1 

24.3 

5S0 

27.8 

28.6 

29.0 

29.4 

29.6 

29.8 

29.8 

29.3 

28.0 

28.7 

27.9 

27,3 

26.6 

600 

255 

26.7 

27.6 

28.4 

28.9 

29.2 

29.6 

29.9 

29.9 

29.8 

29.3 

29,0 

28.5 

620 

22.6 

23.8 

250 

26.2 

27.1 

27.9 

28.8 

29.3 

29.6 

29.8 

30.1 

29.8 

29.6 

640 

19.2 

20.6 

21.6 

23.3 

24.6 

2i.2 

26.6 

27.8 

28.3 

28.9 

29.4 

29.7 

29.9 

660 

162 

17.5 

18.8 

202 

21.1 

22.9 

24.0 

25.1 

26.2 

27.1 

28.2 

28.8 

29.2 

680 

13.8 

14.7 

15.8 

16.9 

18.4 

19.9 

20.6 

22.3 

23.6 

24.9 

25.8 

26.7 

27.5 

700 

11.5 

12.3 

13.4 

14.6 

15.6 

16.7 

18.0 

19.5 

20.7 

22.0 

23.1 

24.2 

25.1 

720 

10.4 

11.0 

11.4 

12.3 

13.3 

14.3 

15.6 

16.4 

17.7 

19.3 

19.9 

21.6 

22.6 

740 

10.3 

10.4 

10.5 

11.0 

11.4 

12.2 

13.3 

14.2 

15.3 

16.5 

17.4 

18.8 

19.5 

760 

10.3 

10.0 

10.2 

10.3 

10.7 

11.0 

11.5 

12.2 

13.1 

14.2 

15.1 

16.0 

17.3 

780 

11.3 

108 

10.6 

10.2 

10.2 

10.5 

10.7 

11.1 

ll.o 

12.3 

13.2 

14.0 

15.0 

800 

13.4 

12.5 

11.7 

11.0 

10.6 

10.3 

10.3 

10.4 

10.7 

11.0 

11.6 

11.3 

12.2 

820 

16.2 

15.2 

14.4 

13.5 

13.5 

11.9 

11.4 

11.0 

10.9 

10.8 

10.8 

11.2 

11.4 

840 

18.3 

17.1 

16.2 

14.9 

14.1 

13.0 

12.4 

11.7 

11.2 

10.7 

10.6 

11.1 

11.2 

860 

21.0 

20.2 

18.7 

17.7 

16.6 

15.4 

14.3 

13.3 

12.5 

11.9 

11.4 

11.0 

10.9 

880 

23.5 

22.4 

21.3 

20.4 

19.3  18.0 

17.0 

15.9 

14.8 

13.7 

12.8 

12,0 

12.6 

900 

24.5 

24.2 

23.8 

22.7 

21.9  19.9 

19.7 

18.6 

17.2 

16.4 

15.3 

14.1 

13.3 

920 

24.5 

24.8 

24.7 

24.3 

24.1  23.2 

22.3 

21.3 

20.0 

19.3 

18.0 

16.7 

15.7 

940 

23.2 

24.0 

24.5 

24.6 

24.'i  24.5 

24.2 

23.5 

92  7 

21.8 

20.6 

19.5 

18.4 

9G0 

20.7 

21.9 

22.8 

23.6 

24.0  24.5 

24.5 

24.2 

24.3 

23.7 

229 

22.1 

21.0 

980 

17.6 

187 

20.1 

21.2 

22.2  23.1 

23.0 

24.0 

24.3 

24.3 

24.3  23.7 

23.0 

1000 

14.3 

15.5 

16.9 

18.2 

19.2  20.2 

21.4 

22.5 

23.0 

23.5 

24.0 

24.2 

24.2 

840 

8T0 

863 

870  1  880  890 

SOD 

910 

920 

930 

940  i  950 

960 

30 


TABLE  XXX.    XXXI. 


Perturbations  by 

Venus. 

Perturbation 

s  by 

Mars. 

Arguments  II  and  III. 

Arguments  II  a.. 

\  IV 

III. 

IV 

11. 

960     97C     930     990 

1000 
21.6 

0 
95 

10 
102 

20 
10.8 

30 
11.2 

40 
11.5 

50 

" 
11.7 

60 
11.8 

70 

0 

24.2    23  7 

23  1    22.5 

11.5 

20 

23.6    23.7 

240    23.4 

23  1 

8.3 

9.1 

9.8 

105 

10.9 

11.2 

11.5 

11.6 

40 

21.6    22.4 

22.9    23.5 

23.5 

7.1 

7.9 

8.8 

9.4 

10.0 

10.6 

10.8 

11.2 

60 

19.1    20  1 

20.7   21.5 

22.2 

5.8 

6.7 

7.6 

S.4 

9.1 

9.8 

10  3 

10.5 

80 

16  2    17.3 

18.4    19.7 

20.0 

4.3 

5.3 

64 

7.2 

8.0 

8.9 

93 

9.9 

100 

14.0    14.8 

15.6-16.5 

17.6 

3.3 

4.2 

5.0 

5.9 

6.8 

7.6 

8.4 

9.1 

120 

12.3    129 

13  7    14,3 

15.3 

2.4 

3.1 

3.9 

4.8 

5.6 

6.4 

7.3 

S.O 

140 

11.5    120 

12.6'  12  8 

13.6 

2.1 

2.4 

2.9 

3.8 

4.6 

5.5 

6.3 

7.0 

160 

11.6    lis 

12.1  :  12.3 

12.7 

2.0 

2.2 

24 

2.7 

3.5 

4.4 

5.1 

5.D 

180 

122    122 

12  3    12  5 

127 

1.9 

2.0 

2.3 

2.6 

2.9 

3.4 

3.9 

t.W 

20J 

12.4    12.7 

12.8:  13.1 

13.2 

23 

2.2 

2.2 

2.4 

2.7 

3'J 

3.4 

3.8 

220 

12.3    12.7 

13.0    13.3 

13.5 

3.0 

2.6 

2.5 

2.4 

2.5 

2.7 

3.1 

3.5 

210 

11.5    12.1 

12.41  13  1 

136 

3.7 

3.3 

3.0 

2,9 

2.7 

2.8 

2.9 

«» «i 

260 

104 

11.0 

11.5:  12.2 

12.8 

4.8 

4.1 

3.7 

3.5 

3.1 

3.1 

3.0 

3.1 

28;) 

88 

96 

10.4    107 

11.5 

5.5 

5.1 

4.6 

4.1 

3.8 

3.5 

3.5 

3.4 

3J0 

7.5 

7.9 

8.6 

9.0 

10.1 

6.2 

5.8 

5.6 

5.0 

4.8 

4.2 

3.9 

3.8 

320 

6.7 

6.8 

7.3 

78 

8.3' 

6.9 

6.6 

6.1 

5,9 

5.4 

5.1 

4.7 

43 

340 

6.6 

6.4 

66 

6.7 

6.2 

7.2 

7.1 

6.9 

6.5 

6.2 

5.8 

5.5 

5.1 

380 

7.4 

69 

65 

6.5 

6.5 

7.5 

7.4 

7.1 

7.0 

6.8 

6.4 

6.2 

5.8 

330 

8.9 

8.2 

7.5 

6.9 

6.8 

7.5 

7.6 

7.3 

7.3 

7.2 

7.1 

6.7 

0.5 

400 

10.9 

10.0 

9.0 

8.3 

7.5 

7.3 

7.3 

7.5 

7.4 

7.4 

7.4 

7.1 

7.0 

420 

12.6 

11.6 

10.7 

99 

9.1 

6.9 

7.0 

7.3 

7.4 

7.4 

7.4 

7.3 

7.5 

410 

14.2 

133 

12  5 

11.6 

10.6 

65 

6.8 

6.8 

7.1 

72 

7.3 

7.3 

7.4 

430 

15.7 

14.S 

13  9 

130 

12  1 

62 

6.2 

6.5 

6.7 

6.8 

7.1 

7.1 

7.3 

430 

16.9 

16.3 

155 

14.5 

13.6 

5.8 

5.9 

60 

6.2 

6.4 

6.5 

7.0 

3.9 

.50  J 

18.1 

17.6 

16.G 

15.8 

15.1 

5.3 

5.4 

5.7 

5.8 

6.0 

6.0 

6.3 

G.G 

.520 

20.1 

19.2 

18.1 

17.4 

16.5 

5.1 

5.1 

5.1 

53 

5.4 

5.6 

5.8 

6.0 

540 

22  0 

21.0 

20.2 

19.2 

18.1 

4,7 

4.8 

4.8 

48 

5.0 

5.1 

5.4 

5.5 

56J 

24  3 

235 

22.6 

21.5 

20.6 

4.4 

4.5 

4.6 

46 

4.7 

4.8 

4.8 

5.0 

530 

23.6 

25  7 

24.9 

23.8 

23.0 

4.2 

4.3 

4.4 

43 

4.5 

4.4 

4.4 

4.5 

600 

23.5 

27.8 

27.0 

26.3 

25.4 

4.0 

4.2 

4.3 

4.2 

4.2 

4.2 

4.2 

4.3 

620 

29.0 

29.2 

23.8 

28.2 

27.4 

4.2 

4.0 

4.1 

4.0 

4.0 

4.0 

4.0 

39 

610 

29.9 

30.0 

29.9 

23.5 

295 

4.3 

4.2 

4.1 

4.0 

4.1 

4.0 

3.9 

3.9 

660 

2. J.  2 

23.5 

29.7 

29.8 

29.9 

4.6 

4.4 

4.3 

4.1 

4.1 

4.1 

4.0 

3.3 

630 

27.5 

23.6 

2S.9 

29.2 

29.7 

4.8 

4.G 

4.5 

43 

4.2 

4.1 

4  0 

3.9 

700 

25.1 

25.4 

27.3 

27.8 

28.7 

5.3 

5.0 

4.8 

4.5 

4.6 

4.0 

4.1 

4.1 

720 

22.6 

23.9 

25.0    28.1 

26.8 

5.8 

5.5 

5.1 

50 

4.7 

4.5 

4.1 

4.1 

740 

19.5 

21.3 

22.5    23.6 

24.6 

65 

6.1 

57 

54 

5.2 

4.9 

4.6 

4.3 

76) 

17.3 

13.6 

19.4    21.0 

22.1 

7.4 

6.7 

64 

6.0 

5.6 

5.3 

5.1 

5.0 

730 

15  0 

15.8 

17.1    18.5 

19.3 

8.2 

7.6 

69 

6.5 

6.4 

5.3 

5.6 

5.3 

8D0 

12.2 

14.1 

14.8  ,  15.9 

17.0 

9.2 

8.5 

8.0 

7.3 

6.8 

6.5 

6.1 

5.8 

823 

11.4 

12.0 

125    13.4 

154 

10.1 

9.6 

88 

82 

7.6 

7.1 

6.7 

6.5 

840 

11.2 

11.3    11.7    12  2 

13.2 

10.9 

10.4 

98 

9.1 

8.4 

7.0 

7.5'    6  9 

SIO 

10.9 

ll>8    JO. 9    11.2 

11.5 

11.7 

11.0 

10.4 

10.0 

9.4 

S.7 

8.2      7.7 

830 

12.6    11.3    11.1    10.8 

11.0 

12.3 

11.9 

11.3 

10  G 

10;: 

9.7 

8.9      S.4 

903 

133    12.3    12.9    11  3 

11.2 

12  4 

12.2 

11.8 

11.6 

lO.S 

10.3 

9.7      9.3 

920 

15.7'  146    13  7    12.8 

12  1 

123 

12.3 

12.2 

11.9 

11.6 

11.0 

10.5      9.9 

040  1  13.4    17  3    1G2    14.5    14  0 

12.1 

12.1 

12  2 

122 

11.8 

11.4 

11.0    10.6 

910    21.0    20  0    18.9    17.9 

10  7 

114 

11.9 

11.9 

12  0 

12.0 

11.7 

11.1    11.0 

933  i  23.0    22  4   21.4   2)3 

19  5 

10.6 

11.1 

11. 0 

US 

ll.C 

;  !.'j 

11.7    11.4 

1000i24.2,23  7   23.1    22  5 

21.6 

9.5 

10  2 

10  3 

11.2 

11  5 

;i.7 

11.8    11.5 

l9G0  i  970     93)     93 J  IlOUO 

0 

10 

23 

30 

40 

.03 

60       70 

TABLE  XXXI. 


31 


Perturbations  inoduced  by  Mars. 

Arguments  II  and  IV. 

IV. 


II. 

70 

80 

90 

100 

110 

120 

^  130 

1  140 

150 

1  160 

170 

1  180 

190 

200 

0 

11.5 

11.2 

11.0 

10.6 

10.1 

9.9 

9.5 

9.0 

8.6 

8.2 

8.1 

7.8 

7.6 

74 

2n 

11.6 

11.4 

11.0 

10.9 

10.6 

10.2 

9.7 

9.1 

9.1 

88 

8.4 

8.1 

7.9 

78 

40 

11.2 

11.3 

11.2 

11.0 

10.8 

10.5 

'  10.3 

9.8 

9.4 

93 

9.1 

8.7 

8.4 

8  2 

60 

10.5 

10.9 

11.1 

10.9 

11.0 

10.9 

10.4 

10.0 

9.7 

95 

9.2 

8.8 

8.7 

8.4 

80 

9.9 

10.0 

10.5 

10.9 

10.8 

10.7 

10.4 

10.3 

10.0 

9.7 

9.3 

9.0 

8.8 

8.6 

100 

9.1 

9.5 

9.8 

10.1 

10.6 

10.5 

10.4 

10.3 

10.1 

9.9 

9.6 

9.3 

9.0 

8.8 

120 

SO 

8.S 

9.3 

95 

9.9 

10.2 

10.2 

10.1 

10.0 

9.8 

9.6 

9.4 

9.1 

89 

140 

7.0 

7.:/ 

8.4 

9.0 

9.3 

9.6 

9.9 

99 

99 

9.7 

9.7 

9.4 

93 

8.9 

160 

5.9 

6.5 

7.2 

80 

8.5 

8.9 

9.2 

9.6 

9.5 

9.6 

9.5 

9.5 

93 

9.1 

ISO 

4.9 

5.6 

6.4 

6.9 

7.7 

8.3 

8.6 

89 

9.4 

9.3 

9.3 

9.3 

9.2 

9.1 

200 

38 

4.6 

5.3 

6.0 

6.7 

7.4 

7.9 

8.3 

8.0 

8.9 

9.1 

9.0 

9.0 

8.9 

220 

3.5 

3  9 

4.4 

.5.1 

5.8 

6.4 

7.1 

7.6 

7.9 

8.4 

8.6 

88 

8.8 

87 

240 

3.2 

3.6 

4.0 

4.4 

5.0 

5.5 

6.2 

6.8 

7.4 

7.6 

8.1 

8.4 

8.4 

8.5 

260 

3.1 

3.2 

3.8 

4.1 

4.5 

4.9 

5.4 

5.9 

6.6 

7.1 

7.5 

7.7 

SO 

8.2 

2S0 

34 

34 

3.5 

3.8 

4.2 

4.5 

4.9 

5.5 

5.6 

6.2 

0.8 

7.1 

75 

7  8 

300 

3.8 

.3.7 

3.7 

3.7 

3.9 

4.4 

4.7 

4.9 

5.4 

5.7 

6.0 

0.6 

0.9 

7  3 

320 

4.3 

4.2 

4.1 

4.0 

4.1 

4.2 

4.4 

4.7 

5.0 

5.4 

58 

6.0 

6.4 

6  6 

340 

5.1 

4.9 

4.6 

4.4 

4.4 

4.3 

4  5 

4.5 

5.0 

5.2 

5  5 

5.8 

6.0 

6  3 

360 

58 

5.6 

5.3 

5.0 

4.8 

4.8 

4.7 

4.8 

4.9 

5.1 

5.4 

55 

59 

6  1 

380 

65 

6.4 

5.9 

5.7 

5.5 

5.4 

5.1 

5.1 

5.1 

5.1 

5.4 

5  5 

5.7 

58 

400 

7.0 

6.7 

6.7 

6.3 

6.1 

5.9 

57 

5.6 

5.5 

5.5 

5.5 

5.0 

5.7 

5.9 

420 

7.4 

7.2 

6.9 

7.1 

6.7 

6.4 

6.3 

6.1 

6.0 

5.9 

59 

5.S 

5.8 

6.1 

440 

7.5 

7.4 

7.4 

7.0 

*  7.1 

7.4 

6.8 

6.7 

6.5 

6.3 

6.3 

64 

0.2 

6.3 

460 

7.3 

7.4 

7.4 

7.5 

7.4 

7.3 

7.3 

7.2 

7.1 

7.1 

6.7 

6.7 

0.7 

6.7 

480 

6.9 

7.1 

7.3 

7.4 

7.5 

7.3 

7.6 

7.5 

7.4 

7.5 

7.4 

7.2 

7  1 

7  1 

500 

6.6 

6.8 

6.9 

7.2 

7.3 

7.5 

7.5 

7.6 

7.8 

7.7 

7.8 

7.7 

7.6 

7.4 

520 

6.0 

6.3 

6.5 

6.7 

7.1 

7.2 

7.5 

7.5 

7.7 

7.8 

7.9 

7.6 

7.9 

7.') 

540 

5.5 

5.7 

6.0 

6.3 

6.6 

6.9 

7.1 

73 

7.4 

7.7 

7.9 

8.0 

8.2 

8.3 

560 

5.0 

5.2 

5.4 

5.8 

5.9 

6.2 

6.6 

6.9 

7.1 

7.4 

7.7 

7.8 

8.1 

8.2 

580 

4.5 

4.7 

4.9 

5.0 

5.3 

5.7 

6.0 

6.6 

6.8 

7.1 

7.2 

7.5 

7.9 

8.2 

600 

4.3 

4.3 

4.4 

4.6 

4.6 

5.0 

6.3 

56 

5.9 

6.5 

6.9 

7.0 

7.4 

7.7 

620 

3.9 

4.0 

40 

4.1 

4.3 

4.4 

4  6 

4.9 

5.3 

5.4 

6.1 

6.6 

6.9 

74 

640 

3.9 

3.8 

3.8 

3.8 

3.9 

3.9 

4.1 

4.3 

4.5 

5.0 

5.2 

5.8 

63 

6.7 

660 

3.8 

3.7 

37 

3.6 

36 

3.7 

3.8 

3.9 

4.1 

4.2 

4.5 

5.0 

53 

6.0 

6S0 

3.9 

3.8 

3.6 

3.4 

3.5 

3.4 

35 

35 

3.6 

3.7 

3.8 

4.2 

4.6 

49 

700 

4.1 

3.9 

3.8 

3.6 

3.5 

3.3 

33 

3.2 

3.2 

3.2 

3.5 

3.6 

38 

4.2 

720 

4.1 

4.1 

4.0 

3.8 

3.6 

35 

3.3 

3.2 

3.3 

3.2 

3.0 

32 

3.4 

3.6 

740 

4.3 

43 

4.2 

4.0 

3.8 

3.7 

3.5 

3.2 

3.0 

30 

2.9 

2.8 

2.9 

3  1 

760 

5.0 

4.7 

44 

4.3 

4.1 

3.8 

37 

3.4 

3.1 

3.0 

2.9 

2.7 

2.7 

2.8 

780 

5.3 

5.1 

4.7 

4.6 

4.4 

4.4 

4.0 

3.8 

3.4 

3.2 

2.9 

2.S 

2.7 

2  5 

800 

5.8 

5.5 

5.4 

4.8 

4.7 

4.7 

4.5 

4.2 

3.9 

3.5 

3.3 

2.9 

2.8 

2.7 

820 

6.5 

6.1 

5.8 

5.6 

5.0 

50 

4.9 

4.6 

4.3 

4.1 

3.6 

3.3 

30 

29 

840 

6.9 

6.7 

6.3 

6.1 

5.8 

5.3 

5.2 

4.9 

4.9 

4.5 

4.2 

:  .9 

3.5 

3' 

860 

7.7 

7.4 

6.9 

6.6 

6.2 

6.2 

5.5 

5.4 

5.2 

5.0 

4.8 

44 

4.1 

3  6 

880 

8.4 

7.9 

7.6 

7.1 

6.9' 

6.4 

6.4 

5.8 

5.7 

5.4 

52 

,':  0 

4.6 

4  3 

900 

9.3 

8.7 

8.3 

7.7 

7.4 1 

^•^1 

0.7 

6.6 

6.1 

6.0 

5.6 

5.4 

5.2 

4.!*, 

920 

9.9 

9.3 

8.8 

8.4 

7.9! 

7.7  1 

7.3 

69 

6.6 

6.3 

6.2 

6.i 

5.6 

6  4 

940 

10.6 

10.1 

9.5 

8.9 

8.7 

8.2 

7.8 

7.6 

7.2 

7.1 

6.5 

6.5 

63 

fi!» 

960 

11.0 

107 

10.3 

9.7 

9.1 

8.7 

8.4 

8.0 

7.8 

7.4 

72 

69 

6.7 

C5 

9S0 

11  4 

11.0 

10,6 

10.2 

9.8 

9.2 

8.9 

8.4 

S.l 

8.0 

7.6 

7.3 

7.2 

69 

1000 

11.5 

11.2 

11.0 

10.6 

10.0 

9.9 

9.5 

9.0 

8.6 

8.2 

8.1 

•.4 

7.6 

7.4 

! 

70 

80 

90 

100 

110  1 

120 

130  ! 

14 

150 

;60 

170 

180  1 

190 

200 

32 


TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II.  and  IV". 

IV. 


II. 

0 

200 
74 

210 
7.2 

220 

230 
6.6 

240 

250 

260 

270 

f  280 

290 
4.7 

300 
4.1 

310 
3.8 

320 
3.4 

7.0 

6.4 

6.2 

5.7 

5.3 

4.9 

20 

78 

7.2 

7.3 

7.2 

7.0 

6.6 

6.3 

6.0 

5.7 

5.3 

5.0 

4.4 

39 

40 

8.2 

8.1 

7.6 

7.5 

7.3 

7.2 

6.8 

6.6 

6.2 

5.9 

5.6 

52 

4.7 

60 

8.4 

8.0 

7.9 

7.8 

7.6 

7.5 

7.3 

7.1 

6.8 

64 

61 

58 

5  4 

80 

8.6 

8.5 

8.2 

8.0 

7.6 

7.7 

7.6 

7.4 

7.1 

7.0 

6.7 

63 

60 

100 

8.8 

8.5 

8.6 

8.4 

8.2 

7.6 

7.7 

7.8 

7.6 

7.3 

7.2 

6.9 

6.6 

120 

8.9 

8.7 

8.4 

8.4 

8.3 

8.3 

8.0 

7.9 

7.7 

7.6 

7.5 

7.3 

7.0 

140 

8.9 

8.7 

8.4 

8.3 

8.2 

8.1 

83 

8.0 

7.9 

7.8 

7.7 

7,5 

7.4 

160 

9.1 

8.9 

8.7 

8.4 

8.3 

8.3 

8.2 

8.1 

8.0 

7.9 

7.9 

7.7 

7.6 

180 

9.1 

8.8 

8.7 

8.5 

8.4 

8.2 

8.0 

8.0 

8.1 

79 

7.8 

80 

7.8 

200 

8.9 

8.8 

8.6 

8.4 

8.4 

8.3 

8.1 

8.0 

7.9 

7.8 

7.8 

7.9 

7.9 

220 

8.7 

8.7 

8.6 

8.4 

8.2 

8.1 

8.0 

7.9 

7.8 

7.7 

7.7 

7.6 

7.7 

240 

8.5 

8.4 

8.5 

8.3 

8.1 

8.0 

7.8 

7.8 

7.8 

78 

78 

7.8 

76 

260 

8.2 

8.2 

8.1 

8.1 

8.1 

7.8 

7.8 

7.7 

7.6 

7.6 

7.6 

7.5 

7.4 

280 

7.8 

7.8 

8.0 

7.8 

7.9 

7.9 

7.7 

7.5 

7.5 

7.3 

7.3 

74 

7.3 

300 

7.3 

7.6 

7.5 

7.6 

7.7 

7.6 

7.6 

7.6 

7.4 

7.3 

7.1 

7.0 

7.1 

320 

6.6 

7.1 

7.3 

7.4 

7.4 

7.3 

7.4 

7.4 

73 

7.1 

7.0 

7.0 

68 

340 

6.3 

64 

67 

7.2 

7.1 

7.2 

7.2 

7.1 

7.1 

70 

6.9 

68 

68 

360 

6.1 

6.2 

6.4 

6.5 

69 

6.9 

7.0 

7.0 

6.9 

6.8 

6.7 

66 

65 

380 

58 

6.1 

6.3 

6.4 

6.6 

6.7 

6.6 

6.6 

6.7 

6.8 

6.7 

6.6 

6.5 

400 

5.9 

6.0 

6.2 

0.3 

6.4 

6.5 

6.6 

6.6 

6.5 

6.6 

6.6 

6.5 

6.4 

420 

6.1 

6.3 

6.2 

6.4 

63 

6.4 

65 

6.6 

6.5 

6,5 

6.5 

65 

64 

440 

6.3 

64 

6.4 

6.6 

6.5 

6.6 

C.5 

65 

6.5 

6.5 

6.3 

63 

6.2 

460 

6.7 

6.5 

6.5 

6.6 

6.7 

6.9 

6.7 

66 

66 

6.6 

6.5 

63 

6.2 

480 

7.1 

7.1 

7.0 

6.9 

69 

6.9 

7.0 

7.0 

6.8 

6.7 

6.6 

6.5 

6.3 

500 

7.4 

7.5 

7.4 

7.4 

7.3 

7.2 

7.3 

7.2 

7.1 

6.9 

6.8 

6.8 

6.6 

520 

7.9 

7.8 

7.8 

7.8 

7.8 

7.6 

7.6 

7.5 

7.5 

7.4 

7.1 

7.0 

6.9 

540 

8.3 

8.3 

8.3 

8.2 

8.2 

8.1 

8.0 

7.9 

7.9 

7.8 

7.6 

7.5 

7.2 

560 

8.2 

8.6 

8.4 

8.6 

8.7 

8.5 

8.5 

8.4 

8.2 

8.3 

8.2 

80 

7.6 

580 

8.2 

8.3 

86 

8.8 

8.8 

9.0 

8.9 

8.9 

8.7 

8.7 

86 

84 

8.4 

600 

7.7 

8.1 

8.5 

8.6 

8.9 

9.1 

9.1 

9.2 

9.2 

9.1 

9.0 

8.8 

8.7 

620 

7.4 

7.6 

8.0 

8.5 

8.7 

9.0 

9.2 

9.5 

9.5 

9.5 

9.4 

93 

9.2 

640 

6.7 

7,2 

7.5 

7.9 

8.3 

8.7 

9.0 

9.3 

9.5 

9.8 

9.8 

9.7 

97 

660 

6.0 

6.3 

7.0 

7.3 

7.7 

8.2 

8.7 

9.0 

9.4 

9.7 

9.8 

10.1 

100 

680 

4.9 

5.6 

6.0 

6.6 

7.1 

7.7 

8.1 

8.5 

9.0 

9.3 

9.8 

100 

10.2 

700 

4.2 

4.5 

5.2 

5.8 

6.4 

6.8 

7.4 

8.0 

8.5 

8.9 

9.2 

9.8 

10.1 

720 

3.6 

39 

4.3 

4.7 

5.3 

5.9 

6.6 

7.0 

7.8 

8.3 

8.8 

9.1 

9.7 

740 

3.1 

33 

3.6 

3.9 

4.4 

4.8 

5.6 

6.2 

6.9 

7.5 

80 

8.7 

9.2 

760 

2.8 

2.8 

3.0 

3.3 

3.6 

4.0 

4.4 

5.1 

5.8 

6.5 

7.2 

7.8 

8.4 

780 

2.5 

2.6 

2.5 

2.7 

3.1 

3.3 

37 

4.1 

4.8 

5.4 

6.1 

6.9 

7.6 

800 

2.7 

2.5 

2.5 

2.5 

2.5 

2.7 

3.0 

3.4 

3.8 

4.4 

5.0 

5.6 

6.6 

820 

2.9 

2.6 

2.4 

2.3 

2.2 

2.3 

2.6 

2.8 

3.1 

3.4 

4.1 

4.7 

5.4 

840 

3.1 

2.8 

2.6 

2.4 

2.3 

2.2 

2.3 

2.4 

2.6 

2.8 

3.2 

38 

43 

860 

3.6 

33 

30 

2.7 

2.4 

23 

2.1 

2.2 

2.3 

2.5 

2.7 

30 

3.4 

880 

4.3 

3.8 

36 

3.2 

2.8 

2.5 

23 

2.1 

20 

2.2 

2.3 

2.5 

2.6 

900 

4.9 

4.6 

4.2 

3.6 

3.4 

2.9 

2.6 

2.3 

2.2 

2.2 

2.1 

2.2 

2.4 

920 

5.4 

5.1 

4.6 

4.5 

3.9 

3.5 

32 

2.9 

2.6 

2.2 

2.0 

2.1 

2.2 

940 

59 

5.7 

5.3 

4.9 

4.7 

4.3 

3.8 

3.4 

30 

2.7 

2.4 

2.1 

20 

960 

6.5 

6.2 

5  9 

5.5 

5.1 

4.9 

4.5 

4.0 

34 

3.1 

2.8 

2.4 

2.3 

980 

6.9 

6.8 

6.4 

6.1 

5.8 

5.4 

5.1 

4.8 

43 

3.9 

3.5 

30 

2.7 

1000 

7.4 
200 

7.2 
210 

7.0 
220 

6.6 
230 

6.4 
240 

6.2 
250 

5.7 

5.3 

4.9 

4.7 
290 

4.1 

3.8 
310 

3.4 

320 

260 

270 

230 

300 

TABLE  XXXI. 


S3 


Perturbations  -produced  by  Mars. 

Arguments  II.  and  IV. 

IV. 


II. 

0 

320 
3.4 

330 

2.8 

340 

350 

360 

370 

380 

390 

400 

1  410 

420 
3.4 

430 
4.0 

440 
4.5 

2.6 

2.4 

2.2 

2.3 

2.3 

2.5 

2.7 

2.9 

20 

3.9 

3.5 

3.1 

2.7 

2.6 

2.4 

2.4 

2.3 

2.5 

2.7 

3.0 

3.3 

3,8 

40 

4.7 

4.2 

3.9 

3.5 

3.0 

2.8 

2.7 

2.6 

2.5 

2.6 

2.8 

2.9 

32 

60 

5.4 

5.0 

4.6 

4.2 

3.8 

3.4 

3,1 

2.8 

2.8 

2.7 

2.7 

2.7 

3.0 

80 

6.0 

5.7 

5.4 

4.8 

4.4 

4.0 

3.6 

3.4 

3.1 

2.9 

2.9 

2.9 

2,9 

100 

6.6 

6.3 

5.9 

5.6 

5.2 

4.8 

4.3 

4.0 

3.7 

3.5 

3.2 

3.0 

3.0 

120 

7.0 

6.9 

6,4 

6.1 

5.8 

5.3 

5.2 

4.6 

4,3 

4,0 

3.8 

3.6 

34 

140 

7.4 

7.2 

6.9 

6.6 

6.5 

6.1 

5.6 

5,4 

5,0 

4,6 

4.3 

4,0 

3,9 

160 

7.6 

7.5 

7.3 

7.0 

6.8 

6.6 

6.2 

5,9 

5,5 

5,3 

4.9 

4,6 

4,4 

180 

7.8 

7.7 

7.5 

7.4 

7.3 

6,9 

6.7 

6,5 

6,2 

5.8 

5.6 

5.3 

50 

200 

7.9 

7.8 

7.7 

7.6 

7.5 

7.3 

7.1 

6.9 

6.6 

6.4 

6.1 

5.6 

5.5 

220 

7.7 

7.7 

7.7 

7.8 

7.7 

7.5 

7.3 

7.2 

7.0 

6.7 

6.5 

6.2 

5.9 

240 

7.6 

7.6 

7.6 

7.6 

7.7 

7.6 

7.5 

7.3 

7.2 

7.1 

6.9 

6.6 

6.4 

260 

7.4 

7.3 

7.5 

7.5 

7.5 

7.6 

7.6 

7.5 

7.5 

7.3 

7.1 

7,0 

6,7 

280 

7.3 

7.4 

7.3 

7.3 

7.4 

7.4 

7.3 

7.4 

7.3 

7.5 

7.2 

7.1 

6.9 

300 

7.1 

7.1 

7.1 

7.0 

7.2 

7.3 

7.3 

7,3 

7.2 

7.2 

7.3 

7.2 

7.1 

320 

6.8 

6.8 

6.9 

6.9 

6.8 

7.0 

7.1 

7.1 

7.1 

7.1 

7.1 

7,0 

7.2 

340 

6.8 

6.7 

6.6 

6.6 

6.6 

6.8 

6.9 

6.9 

7.0 

7.0 

6.9 

69 

6.9 

360 

0.5 

6.5 

6.4 

6.3 

6.4 

6.5 

6.6 

6.7 

6.8 

6.8 

6.8 

6,8 

6.9 

380 

6.5 

6.3 

6.3 

6.2 

6.2 

6.2 

6.3 

6,3 

6.4 

6,5 

6.6 

6.7 

6.7 

400 

6.4 

6.2 

6.2 

6.0 

6.1 

6.0 

6.0 

6.0 

6.0 

6.1 

6.2 

6.3 

6.4 

420 

6.4 

6.2 

6.1 

6.0 

5.9 

5.8 

5.9 

5.9 

5.9 

5.9 

5,9 

6.0 

6.0 

440 

6.2 

6.1 

6.0 

5.8 

5.8 

5.7 

5.6 

5,6 

5.6 

5.7 

5,7 

5.8 

5.9 

460 

6.2 

6.0 

5.9 

5.8 

5.7 

5,5 

5,5 

5,4 

5.5 

5.4 

5.5 

53 

5,4 

480 

63 

6.2 

6.0 

5.7 

5,6 

5.5 

5.4 

5.3 

5.2 

5,2 

5.2 

5,3 

5,3 

500 

6.6 

6.4 

6.2 

6.0 

5.7 

5.4 

5.3 

5.2 

5.1 

5.1 

5.1 

5.0 

5.0 

520 

6.9 

6.7 

6.4 

6.1 

6.1 

5.7 

5.5 

5.1 

5.1 

5.0 

4.9 

5.0 

4.9 

540 

7.2 

7.1 

6.7 

6.5 

6.2 

6.1 

5.8 

5,5 

5.2 

5,0 

4.9 

4,8 

4.8 

560 

7.6 

7.4 

7.3 

7.0 

6.6 

6.3 

6,0 

5.8 

5.4 

5,3 

5.0 

4,7 

4.7 

580 

8-4 

8.0 

7.8 

7.5 

7.0 

6.8 

6,3 

6,2 

5.9 

5,5 

5.3 

5.0 

4.9 

600 

8.7 

8.6 

8.3 

8.0 

7.8 

7.4 

7.0 

6.6 

6.3 

6.0 

5.6 

5.3 

5.1 

620 

9.2 

9.1 

8.9 

8.6 

8.4 

8.1 

7.6 

7.2 

6.8 

6,5 

6.1 

5.7 

5.3 

040 

9.7 

9.0 

9.4 

9.3 

9.0 

8.7 

8.2 

7.8 

7.4 

7,0 

6,6 

6,3 

5.8 

660 

10.0 

10.0 

9.9 

9.8 

9.6 

9.3 

8.9 

8.5 

8,2 

7,7 

7,2 

6.8 

6.4 

680 

10.2 

10.4 

10.3 

10.2 

10.1 

9.9 

9.6 

9.3 

9.0 

8.5 

8.1 

7.5 

7.1 

700 

10.1 

10.3 

10.5 

10.6 

10.4 

10.3 

10.1 

9.8 

9.6 

9.3 

8.9 

8.3 

7.8 

720 

9.7 

10.1 

10.3 

10.6 

10.7 

10.6 

10,5 

10,5 

10,2 

10.0 

9.6 

9,2 

8.6 

740 

9.2 

9.6 

10.0 

10.3 

10.6 

10.7 

10,8 

10.9 

10,6 

:o,5 

10.2 

9,9 

9,4 

760 

8.4 

9.0 

9.5 

9.8 

10.2 

10.6 

10.9 

11.0 

11,0 

11,0 

10,7 

10,5 

10.3 

780 

7.6 

8.2 

8.9 

9.4 

9.9 

10,3 

10,6 

10,9 

11,1 

11,2 

11.0 

10.8 

10.7 

800 

6.6 

7.3 

7.9 

8.5 

9.2 

9.8 

10.1 

10.6 

10,8 

11.1 

11.3 

11.1 

11.0 

820 

5.4 

6.0 

7.0 

7.6 

8.2 

8,9 

9,6 

10.0 

10,5 

10.8 

11.0 

11,3 

11.3 

840 

4.3 

5.0 

5.6 

6.5 

7.2 

7.9 

9.8 

9,2 

9,9 

10.3 

10.7 

10.9 

11.2 

860 

34 

4.0 

4.6 

5.3 

6.1 

6.9 

7.5 

8.4 

9.1 

9.6 

10.1 

10,7 

10.9 

880 

2.6 

3.1 

3.7 

4.3 

5.0 

5.7 

6.6 

7.1 

8.1 

8.7 

9.4 

9,8 

10.4 

900 

2.4 

2.7 

3.0 

3.4 

4.0 

4.6 

5.4 

6.1 

6.9 

7.6 

8.4 

9.1 

9.7 

920 

2.2 

2.3 

2.3 

2.8 

3.3 

3.7 

4.3 

5.0 

5.8 

6.5 

7.2 

8.0 

8.7 

940 

2.0 

2.1 

2.3 

2.3 

2.7 

2.9 

3.4 

4.1 

4.7 

5.5 

6.1 

7.0 

7.7 

960 

2.3 

2.2 

2.2 

2.3 

2.3 

2.5 

2.8 

3.2 

39 

4.5 

5.1 

b.1 

6.5 

980 

2.7 

2.4 

2.2 

2.3 

2.3 

2.4 

2,5 

2.8 

3.0 

3.6 

4.1 

4.7 

5.5 

1000 

3.4 

2.8 

2.6 
340 

2.4 

a50 

2.2 

2.3 

2.3 

2.5 
390 

27 
400 

2.9 
410 

3.4 
420 

4.0 

430 

4.5 
440 

320 

330 

360 

370 

380 

34 


TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II  and  IV. 

IV. 


IT. 
0 

440 

450 

460 
5.9 

470 
6.6 

480 
7.3 

490 

500 

510 
9.0 

520 
9.5 

530 
10.0 

540 

550 
10.7 

.560 

4.5 

5.2 

8.0 

8.5 

10.4 

10.9 

20 

3.8 

4.3 

4.9 

5.6 

6.2 

6.9 

7.6 

8.2 

8.8 

9.3 

9.7 

10.0 

11.4 

40 

3.2 

3.7 

4.2 

4.8 

5.4 

5.9 

6.6 

7.3 

7.9 

8.4 

8.9 

94 

9.8 

60 

3.0 

3.2 

3.6 

4.0 

4.5 

5.1 

5.7 

63 

6.9 

7.5 

8.0 

8.6 

9.1 

80 

2.9 

3.1 

33 

3.5 

3.9 

4.4 

4.9 

5.4 

5.9 

6.5 

7.1 

7.7 

8.2 

100 

3.0 

3.1 

3.2 

3.5 

3.6 

3.8 

4.2 

4.8 

5.3 

5.9 

6.4 

6.9 

7.4 

120 

3.4 

3.3 

3.3 

3.4 

3.5 

3.6 

3.9 

4.2 

4.7 

5.1 

5.6 

6.0 

6.6 

140 

3.9 

3.8 

3.6 

3.6 

3.6 

3.7 

4.0 

4.0 

4.2 

4.6 

5.0 

5.4 

5.9 

160 

4.4 

4.2 

3.9 

4.1 

3.8 

3.7 

4.0 

4.1 

4.2 

4.5 

4.6 

4.9 

5.3 

180 

5.0 

4.8 

4.4 

4.2 

4.2 

4.2 

4.0 

4.1 

4.3 

4.4 

4.4 

4.7 

5.0 

200 

5.5 

5.2 

5.1 

4.8 

4.6 

4.5 

4.5 

4.4 

4.5 

4.5 

4.7 

4.6 

4.8 

220 

5.9 

5.7 

5.5 

5.3 

5.1 

4.9 

4.9 

4.8 

4.7 

4.8 

4.8 

4.9 

5.0 

240 

6.4 

6.2 

5.9 

5.8 

5.6 

5.4 

5.3 

5.2 

5.1 

5.1 

5.1 

5.2 

5.2 

260 

6.7 

6.6 

6.4 

6.1 

6.0 

5.9 

5.8 

5.7 

5.6 

5.5 

5.4 

5.4 

5.4 

280 

6.9 

6.8 

6.7 

6.5 

6.3 

6.2 

6.1 

6.0 

5.9 

5.9 

5.9 

5.8 

5.8 

300 

7.1 

7.0 

6.8 

6.8 

6.6 

6.5 

6.4 

6.3 

6.2 

6.2 

6.2 

6.2 

6.2 

320 

7.2 

7.1 

6.9 

6.8 

6.8 

6.7 

6.6 

6.5 

6.5 

6.5 

6.5 

6.6 

6.6 

340 

6.9 

6.9 

7.0 

6.9 

6.9 

6.8 

6.7 

6.8 

6.7 

6.6 

6.7 

6.8 

6.9 

360 

6.9 

6.8 

6.8 

6.8 

6.8 

6.7 

6.7 

6.6 

6.6 

6.8 

6.8 

6.8 

6.9 

380 

6.7 

6.5 

6.5 

6.6 

6.7 

6.6 

6.6 

6.7 

6.7 

6.7 

6.8 

6.9 

6.9 

400 

6.4 

6.4 

6.3 

6.3 

6.4 

6.5 

6.5 

6.5 

6.6 

6.7 

6.7 

6.8 

6.8 

420 

6.0 

6.2 

6.3 

6.3 

6.2 

6.2 

6.3 

6.3 

6.3 

6.3 

6.5 

6.6 

6.7 

440 

5.9 

5.9 

6.0 

6.0 

6.0 

6.0 

6.0 

6.1 

6.0 

6.1 

6.2 

6.2 

6.4 

460 

5.4 

5.5 

5.7 

5.8 

5.8 

5.8 

5.8 

5.8 

5.8 

5.8 

5.9 

6.0 

6.1 

480 

5.3 

5.3 

5.5 

5.5 

5.5 

5.6 

5.5 

5.6 

5.4 

5.6 

5.7 

5.5 

5.8 

500 

5.0 

5.0 

5.1 

5.2 

5.3 

5.3 

5.3 

5.2 

5.2 

5.2 

5.3 

5.4 

5.4 

520 

4.9 

4.9 

4.9 

4.8 

5.0 

5.1 

5.1 

5.1 

5.1 

5.1 

5.0 

5.0 

5.1 

540 

4.8 

4.8 

4.7 

4.8 

4.8 

4.9 

4.9 

5.0 

4.9 

4.8 

4.8 

4.9 

4.8 

560 

4.7 

4.6 

4.6 

4.7 

4.7 

4.6 

4.7 

4.7 

4.7 

4.7 

4.6 

4.6 

4.6 

580 

4.9 

4.6 

4.5 

4.5 

4.6 

4.5 

4.4 

4.4 

4.5 

4.5 

4.5 

4.4 

4.4 

600 

5.1 

4.9 

4.6 

4.5 

4.4 

4.4 

4.4 

4.3 

4.3 

4.3 

4.3 

4.3 

4.3 

620 

5.3 

5.1 

4.9 

4.7 

4.6 

4.4 

4.3 

4.1 

4.2 

4.2 

4.2 

4.2 

4.1 

640 

5.8 

5.4 

5.2 

5.0 

4.7 

4.6 

4.4 

4.1 

4.1 

4.1 

4.2 

4.2 

4.0 

660 

6.4 

6.0 

5.7 

5.4 

5.0 

4.8 

4.7 

4.5 

4.3 

4.2 

4.2 

4.1 

4.0 

680 

7.1 

6.6 

6.2 

5.7 

5.4 

5.1 

4.9 

4.7 

4.5 

4.4 

4.3 

4.0 

3.9 

700 

7.8 

7.2 

6.8 

6.4 

6.0 

5.6 

5.3 

5.0 

4.7 

4.6 

4.6 

4.3 

4.1 

720 

86 

8.0 

7.6 

7.1 

6.6 

6.2 

5.7 

5.5 

5.2 

4.9 

4.6 

4.6 

4.3 

740 

9.4 

9.0 

8.4 

8.0 

7.4 

6.9 

6.3 

6.0 

5.6 

5.3 

5.0 

4.7 

4.5 

760 

10.3 

9.7 

9.3 

8.6 

8.1 

7.6 

7.2 

6.5 

6.2 

5.8 

5.5 

5.2 

4.9 

780 

10.7 

10.5 

9.9 

9.6 

9.0 

8.5 

7.8 

7.4 

7.0 

6.4 

6.1 

5.7 

5.5 

800 

11.0 

11.0 

10.6 

10.2 

9.9 

9.3 

8.8 

8.1 

7.7 

7.3 

6.7 

6.3 

5.8 

820 

11.3 

11.1 

10.9 

10.6 

10.3 

10.0 

9.6 

9.1 

8.5 

7.9 

7.4 

7.0 

6.6 

840 

11.2 

11.3 

11.2 

11.1 

11.0 

10.7 

10.2 

9.9 

9.4 

8.8 

8.2 

7.7 

7.3 

860 

10.9 

11.1 

11.4 

11.3 

11.3 

11.2 

10.7 

10.4 

9.9 

9.6 

9.2 

8.5 

7.9 

880 

10.4 

10.8 

11.0 

11.3 

11.2 

11.2 

11.2 

10.9 

10.5 

10.3 

9.8 

9.3 

8.7 

900 

9.7 

10.1 

10.6 

11.0 

11.2 

11.2 

11.2 

11.0 

10.9 

10.7 

10.2 

10.0 

9.4 

920 

8.7 

9.3 

9.9 

10.3 

10.8 

11.0 

11.1 

11.2 

11.2 

11.0 

10.7 

10.4 

10.1 

940 

7.7 

82 

8.8 

9.5 

10.1 

10.4 

10.9 

11.0 

11.2 

11.2 

11.0 

10.7 

10.5 

960 

6.5 

73 

8.1 

8.6 

9.3 

9.8 

10.2 

10.6 

10.8 

11.1 

11.2 

10.9 

10.8 

930 

5.5 

62 

7.0 

7.7 

8.3 

8.9 

9.5 

10.0 

10.4 

10.6 

10.8 

110 

10.9 

1000 

4.5 
440 

5.2 
450 

5.9 
460 

6.6 
470 

7.3 

8.0 
490 

8.5 
500 

9.0 
510 

9.5 
520 

10  0 
530 

10.4 

10.7 
550 

10.9 

480 

540 

560 

TABLE  XXXI. 


35 


Perturbations  produced  by  Mars. 

Arguments  II  and  IV. 

IV. 


II. 

560 

570 

580 

590 
10,4 

600 

610 

620 

630 

640 
8.9 

650 

660 

670 

680 
7.7 

0 

10  9 

10.8 

10.6 

10.3 

10.0 

9.7 

9,2 

8.5 

8.1 

7.9 

20 

11.4 

10.6 

10.7 

10,6 

10.4 

10.2 

9.9 

9.7 

9.3 

9.0 

8.8 

8.5 

8.1 

40 

9.8 

10.1 

10.4 

10,4 

10.5 

10.3 

10.2 

9,9 

9.6 

9.4 

9.1 

8.9 

8.5 

60 

9.1 

9.4 

9.8 

10,2 

10.2 

10.3 

10.2 

10.1 

9.9 

9.6 

9.3 

9.0 

8.8 

80 

8.2 

8.7 

9.0 

9.3 

9.6 

9.8 

10.0 

9.9 

9.8 

9.7 

9.5 

9.3 

9.1 

100 

7.4 

7.9 

8.4 

8.7 

9.0 

9.4 

9.6 

9.7 

9.8 

9.7 

9.7 

9.5 

9.2 

120 

6.6 

6.9 

7.6 

8.1 

8.3 

8.6 

9.0 

9.2 

9.4 

9.5 

9.5 

9.4 

9.3 

140 

5.9 

6.3 

6.8 

7.2 

7.7 

8.0 

8.3 

8.7 

8.9 

9.1 

9.2 

9.3 

9.3 

160 

5.3 

5.8 

6.0 

6.5 

6.9 

7,4 

7.7 

8.0 

8.4 

8.5 

8.8 

8.9 

9.0 

180 

5.0 

5.2 

5.6 

6.0 

6.3 

6.7 

7.1 

7.2 

7.7 

8.1 

8.1 

8.4 

8.6 

200 

4,8 

5.0 

5.3 

5.4 

5.8 

6.1 

6.5 

6.7 

7.1 

7.3 

7.7 

7.8 

8.0 

220 

5.0 

5.0 

5.1 

5.3 

5.5 

5.7 

6.0 

6.3 

6.6 

6.8 

7.0 

7.3 

7.5 

240 

5.2 

5.2 

5.3 

5.3 

5.4 

5.5 

5.7 

5,9 

6.1 

6,4 

6.G 

6.8 

7.1 

260 

5.4 

5.5 

5.5 

5.5 

5.5 

5.5 

5.6 

5.7 

5.8 

6,0 

6.3 

6.4 

6.5 

280 

5.8 

5.8 

5.8 

5.9 

5,8 

5.8 

5.8 

5.9 

5.9 

5.9 

6.0 

6.1 

6.2 

300 

6.2 

6.1 

6.2 

6.1 

6.1 

6.1 

6.2 

6.1 

6.0 

5.9 

5.9 

6.0 

6.1 

320 

6.6 

6.5 

6.6 

6.6 

6.5 

6.5 

6.6 

6.5 

6.5 

6.3 

6.1 

6.0 

6.0 

340 

6.9 

6.9 

6.9 

7.0 

7.0 

C.9 

6,8 

6.9 

6.9 

6.8 

6.6 

6.5 

6.3 

360 

6.9 

7.0 

7.2 

7.3 

7.3 

7.3 

7.4 

7.3 

7.3 

7.1 

7.1 

7.0 

6.7 

380 

6.9 

7.0 

7.2 

7.4 

7.5 

7.6 

7.7 

7.7 

7.7 

7.6 

7.5 

7.4 

7.2 

400 

6.8 

7.0 

7.1 

7.3 

7.6 

7.9 

8.0 

8.0 

8.1 

8.1 

8.1 

7.9 

7.8 

420 

6.7 

6.9 

7.0 

7.2 

7.6 

7.8 

8.0 

8.2 

8.3 

8.4 

8.4 

8.5 

8.4 

440 

6.4 

6.6 

6.9 

7.0 

7.3 

7.5 

7.9 

8.2 

8.4 

8.6 

8.8 

8.8 

8.9 

460 

6.1 

6.2 

6.5 

6,9 

7.1 

7.2 

7.6 

8.0 

8.4 

8.7 

9.0 

9.1 

9.2 

480 

5.8 

5.9 

6.0 

6.2 

6.7 

7.1 

7.2 

7.6 

7.9 

8.5 

8.9 

9.2 

9,3 

500 

5.4 

5.5 

5.6 

5.9 

6.1 

6.4 

6.9 

7.2 

7.7 

7.9 

8.4 

9.0 

9.4 

520 

5.1 

5.2 

5.2 

5.3 

5.6 

5.9 

6.3 

6.7 

7.0 

7.6 

8.0 

8.4 

9.0 

540 

4.8 

4.8 

4,8 

5.0 

5.1 

5.4 

5.6 

6.0 

6.4 

6.7 

7.5 

8.1 

8.5 

560 

4.6 

4.5 

4.5 

4.5 

4.7 

4.8 

5.0 

5.3 

5.8 

6.2 

6.6 

7.1 

7.8 

580 

4.4 

4.3 

4.3 

4.3 

4.3 

4.3 

4.5 

4.7 

5.2 

5.5 

5.9 

6.4 

6.9 

600 

4.3 

4.3 

4.2 

4.1 

4.0 

4.0 

4.1 

4.2 

4.5 

4.8 

5.1 

5.7 

6.2 

620 

4.1 

4.0 

4.0 

3,9 

3.9 

3.8 

3.8 

3.8 

3.8 

4.0 

4.4 

4.9 

5.4 

640 

4.0 

3.9 

4.0 

3,8 

3,8 

3.8 

3.7 

3.5 

3.5 

3.6 

3.8 

4.0 

4.5 

660 

4.0 

4.0 

39 

3,8 

3.7 

3.5 

3.5 

3.4 

3.3 

3.3 

3.4 

3.5 

3.7 

680 

3.9 

4.0 

39 

3.8 

3.6 

3.5 

3.4 

3.3 

3.2 

3.1 

3.1 

3.1 

3.1 

700 

4.1 

3.9 

3.9 

3.9 

3.7 

3.5 

3.4 

3.3 

3.2 

3.0 

3.0 

3.0 

2.9 

720 

43 

4.1 

4.0 

3.9 

3.8 

3.8 

3.5 

3.4 

3.1 

2.9 

2.9 

2.7 

2.7 

740 

4.5 

4.2 

4,2 

4.2 

4.0 

3.7 

36 

3.4 

3.3 

3.0 

2.8 

2.6 

2.5 

760 

4.9 

4.7 

4,5 

4.3 

4.2 

4.1 

3.8 

3.6 

3,3 

3.1 

2.9 

2.8 

2.5 

780 

5.5 

5.1 

4,9 

4.5 

4.4 

4.3 

4.1 

3.9 

3.8 

3.4 

3.2 

3.0 

2.7 

800 

5.8 

5.6 

5.2 

5.0 

4.6 

4.5 

4.4 

4.3 

4.1 

3.8 

3.5 

3.1 

2.8 

820 

6.6 

6.1 

5.8 

5.5 

5.3 

5.0 

4,8 

4.6 

4.4 

4.2 

4.0 

3.6 

3.3 

840 

7.3 

6.8 

6.5 

6.1 

5.7 

5.5 

5.2 

5.0 

4.7 

4.6 

4.3 

4.1 

3.8 

860 

7.9 

7.5 

7.0 

6.7 

6.4 

5,9 

5.8 

5.4 

5.1 

5.0 

4.8 

4.6 

4.4 

880 

8.7 

8.2 

7,8 

7.3 

6.9 

6,6 

6.3 

6.0 

5.7 

5.4 

5.2 

5.0 

4.7 

900 

9.4 

9.0 

8.5 

8.0 

7.6 

7.2 

6.8 

6.6 

6.3 

5.9 

5.6 

5.4 

5.2 

920 

10.1 

9.8 

9,2 

8.7 

8.3 

7,8 

7.4 

7.0 

6.7 

6.4 

6.0 

5.8 

5.7 

940 

10.5 

10.2 

9,8 

9.4 

8,8 

8.5 

8.0 

7,6 

7.3 

6,9 

6.6 

6,2 

61 

960 

10.8 

10.5 

10.2 

10,0 

95 

9,1 

8,6 

8.2 

7.8 

7.5 

7.1 

6.8 

66 

980 

109 

10.7 

10.3 

10.2 

9,9 

9,6 

9,2 

9.0 

8,5 

8.0 

7.7 

7.4 

72 

1000 

10.9 

10.8 
570 

10,6 

10.4 
590 

10.3 
600 

10,0 

9,7 

9.2 

8.9 

8.5 

8.1 

7.9 

7.7 

560 

580 

610 

620 

630 

640 

650 

CO 

670  1 

680 

36 


TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  11.  and  IV. 

IV. 


II. 

0 

680 

690 
7.4 

700 
6.9 

710 

720 
6.7 

730 

740 

750 

760 

770 
5.2 

780 

790 

800 

7.7 

6.8 

6.4 

6.1 

5.8 

5.5 

4.8 

4.4 

3.7 

20 

8.1 

7.8 

7.4 

7.0 

7.1 

6.9 

6.7 

6.4 

6.1 

5.8 

5.5 

5.1 

4.7 

40 

8.5 

8.3 

7.8 

7.5 

7.2 

7.1 

7.0 

6.9 

6.6 

6.4 

6.1 

5.8 

53 

60 

8.8 

8.6 

8.3 

8.1 

7.8 

7.6 

7.5 

7.4 

7.1 

6.9 

6.7 

6.3 

6.0 

80 

9.1 

8.9 

8.7 

8.4 

8.1 

8.0 

7.8 

7.6 

7.4 

7.3 

7.1 

6.9 

6.5 

100 

9.2 

8.9 

8.8 

8.7 

8.6 

8.3 

8.0 

7.7 

7.6 

7.6 

7.6 

7.3 

7.0 

120 

9.3 

9.2 

9.0 

8.7 

8.6 

8.4 

8.2 

8.1 

7.9 

7.8 

7.7 

7.6 

7.5 

140 

9.3 

9.2 

9.0 

9.0 

8.7 

8.5 

8.4 

8.3 

8.0 

7.8 

7.7 

7.7 

7.7 

160 

9.0 

9.0 

8.9 

8.8 

8.7 

8.6 

8.5 

8.4 

8.2 

8.0 

7.9 

7.8 

7-8 

180 

8.6 

8.6 

8.7 

8.7 

8.7 

8.6 

8.5 

8.3 

8.3 

8.0 

8.2 

7.8 

7.9 

200 

8.0 

8.2 

8.3 

8.3 

8.5 

8.4 

8.4 

8.4 

8.2 

8,1 

8.1 

8.1 

7.9 

220 

7.5 

7.7 

7.9 

8.1 

8.2 

8.2 

8.1 

8.2 

8.2 

8.0 

8.1 

8.0 

80 

240 

7.1 

7.2 

7.4 

7.5 

7.6 

7.7 

7.8 

7.8 

7.9 

8.0 

8.0 

7,8 

7.8 

260 

6.5 

6.7 

6.9 

7.1 

7.2 

7.3 

7.4 

7.5 

7.6 

7.6 

7.7 

7.7 

7.8 

280 

6.2 

6.3 

6.5 

6.7 

6.7 

6.9 

7.1 

7.2 

7.3 

7.3 

7.3 

7.3 

7.4 

300 

6.1 

6.0 

6.2 

6.4 

6.4 

6.5 

6.6 

6.7 

6.9 

6.9 

6.9 

7.1 

7.1 

320 

6.0 

6.0 

6.0 

6.0 

6.2 

6.1 

6.2 

6.3 

6.5 

6.5 

6.6 

6.6 

6.8 

340 

6.3 

6.2 

6.0 

6.0 

6.0 

6.0 

6.1 

6.1 

6.2 

6.2 

6.3 

6.3 

6.4 

360 

67 

6.6 

6.4 

6.1 

6.0 

5.9 

6.0 

5.9 

59 

5.9 

6.0 

6.1 

6.2' 

380 

7.2 

7.1 

6.8 

6.6 

6.4 

6.2 

6.1 

5.9 

5.8 

5.7 

5.6 

5.8 

5.9 

400 

7.8 

7.7 

7.4 

7.1 

6.8 

6.6 

6.4 

6.1 

6.0 

5.8 

5.6 

5.5 

5.6 

420 

8.4 

8.2 

8.0 

7.8 

7.5 

7.2 

6.8 

6.5 

6.2 

6.0 

5.7 

5.5 

5.4 

440 

8.9 

8.8 

8.7 

8.4 

8.2 

7.8 

7.5 

7.1 

6.6 

6.2 

6.0 

5.7 

5.6 

460 

9.2 

9.2 

9.2 

9.0 

8.8 

8.5 

8.2 

7.9 

7.5 

6.9 

6.5 

6.3 

6.0 

480 

9.3 

9.5 

9.6 

9.6 

9.4 

9.2 

9.1 

8.6 

8.3 

7.8 

7.2 

6.9 

6.5 

500 

9.4 

9.6 

9.8 

10.0 

9.9 

9.8 

9.6 

9.4 

9.1 

8.7 

8.2 

7.6 

7.2 

520 

9.0 

9.5 

9.8 

10.1 

10.2 

10.3 

103 

10.0 

9.8 

9.5 

9.1 

8.5 

8.0 

540 

8.5 

9.1 

9.5 

10.0 

10.3 

10.5 

10.6 

10.6 

10.4 

10.1 

9.8 

9.5 

9.0 

560 

7.8 

8.5 

9.0 

9.5 

9.9 

10.4 

10.8 

10.8 

10.9 

10.8 

10.6 

10.2 

9.9 

580 

6.9 

7.6 

8.3 

90 

9.7 

10.0 

10.4 

10.7 

11.1 

11.2 

11.0 

11.0 

10.6 

600 

6.2 

6.8 

7.4 

8.0 

8.9 

9.6 

10.1 

10.4 

10.9 

11.3 

11.4 

11.3 

11.2 

620 

5.4 

5.9 

6.5 

7.1 

7.8 

8.6 

9.4 

103 

10.6 

11.0 

11.5 

11.7 

11.7 

640 

4.5 

5.0 

5.5 

6.2 

68 

7.6 

8.4 

9.2 

10.0 

10.7 

11.1 

11.6 

11.8 

660 

3.7 

4.1 

4.7 

5.2 

59 

6.5 

7.3 

8.3 

9.1 

9.8 

10.5 

11.2 

11.5 

680 

3.1 

3.4 

3.8 

4.3 

4.8 

5.5 

6.2 

7.0 

7.8 

87 

9.6 

10.2 

11.0 

700 

2.9 

2.8 

3.0 

3.4 

3.9 

4.5 

5.2 

6.0 

6.7 

7.5 

8.5 

9.4 

10.1 

720 

2.7 

2.6 

2.5 

2.7 

3.1 

3.5 

4.0 

4.8 

5.6 

6.4 

7.3 

8.2 

9.1 

740 

2.5 

2.4 

24 

2.4 

2.5 

2.7 

3.1 

3.6 

4.5 

5.2 

6.1 

6.9 

7.8 

760 

2.5 

2.3 

22 

2.1 

2.1 

2.3 

2.4 

2.8 

3.2 

4.1 

4.7 

5.7 

6.6 

780 

2.7 

2.5 

23 

2.1 

2.0 

1.9 

2.1 

2.2 

2.5 

2.9 

3.6 

4.4 

5.2 

800 

2.8 

2.7 

2.4 

2.2 

2.0 

1.8 

1.8 

1.8 

2.0 

2.3 

2.5 

3.2 

4.0 

820 

3.3 

3.0 

2.7 

23 

2.1 

1.9 

1.8 

1.5 

1.7 

1.7 

2.0 

22 

2.9 

840 

3.8 

3.5 

3.0 

2.6 

2.3 

2.1 

1.9 

1.6 

1.5 

1.5 

1.6 

1.7 

2.2 

860 

4.4 

4.0 

3.5 

32 

2.8 

2.3 

1.9 

1.7 

1.4 

1.3 

1.2 

1.4 

1.6 

880 

4.7 

4.4 

4.1 

3.7 

3.3 

3.0 

2.5 

2.1 

1.7 

1.4 

1.3 

1.2 

1.2 

900 

5.2 

5.0 

4.6 

4.3 

4.0 

3.6 

3.2 

2.7 

2.2 

l.G 

1.3 

1.2 

1.1 

920 

5.7 

53 

5.1 

5.0 

4.6 

4.2 

3.8 

3.4 

2.9 

2.3 

1.9 

1.3 

1.1 

940 

6.1 

5.9 

5.6 

54 

52 

4.8 

4.5 

3.9 

3.5 

3.1 

2.6 

2.1 

1.5 

960 

6.6 

6.4 

6.2 

5.9 

5.6 

5.4 

5.1 

47 

43 

3.7 

3.2 

2.8 

2.3 

980 

7.2 

6.9 

66 

6.4 

62 

5.9 

56 

53 

•5.0 

4.6 

4.0 

3.5 

3.0 

1000 

7.7 
680 

y.4 

6.9 
700 

68 

6.7 
720 

6.4 
730 

6.1 
740 

5.8 

5.5 
760 

52 
770 

4.8 
780 

4.4 
790 

3.7 
800 

690 

710 

750 

TABLE  XXXI. 


37 


Perturbations  produced  hy  Mars. 

Arguments  II.  and  IV. 

IV. 


II. 

800 

810 

820 

830 

1  840 

850 

1  860 

870 

880 

890 

900  910 

920 

0 

3.7 

3.2 

2.6 

2.1 

1.7 

1.3 

0.9 

0.7 

0.7 

1.0 

1.2'  1.6 

22 

20 

4.7 

4.2 

3.6 

3.1 

2.4 

1.9 

1.5 

1.2 

0.8 

0.6 

0.9 

1.2 

1  5 

40 

5.3 

4.9 

4.5 

3.8 

3.3 

2.7 

2.0 

1.7 

1.4 

1.0 

0.8 

0.9 

1.0 

60 

6.0 

5.7 

5.2 

4.7 

4.1 

3.6 

3.1 

2.6 

2.0 

1.5 

1.2 

0.9 

1.0 

80 

6.5 

6.3 

6.0 

5.5 

5.0 

4.6 

4.0 

3.4 

2.7 

2.2 

1.8 

,  1-5 

1.3 

100 

7.0 

6.7 

6.5 

6.3 

5.9 

5.3 

4.9 

4.4 

3.7 

3.1 

2.5 

J  2.1 
1  2.9 

1.7 

120 

7.5 

7.3 

7.0 

6.8 

6.5 

6.2 

5.7 

5.1 

4.7 

4.1 

3.5 

24 

140 

7.7 

7.7 

7.5 

7.3 

7.0 

6.7 

6.4 

6.0 

5.6 

5.1 

4.5 

3.8 

33 

160 

7.8 

7.9 

7.7 

7.6 

7.4 

7.2 

7.0 

6.8 

6.3 

5.8 

5.4 

4.8 

42 

180 

7.9 

7.8 

7.9 

7.9 

7.7 

7.6 

7.5 

7.1 

7.0 

6.6 

6.1 

5.7 

5.2 

200 

7.9 

7.9 

7.8 

7.9 

7.8 

7.7 

7.6 

7.5 

7.5 

7.1 

6.8 

6.3 

6.1 

220 

8.0 

7.9 

7.8 

7.8 

7.8 

7.8 

7.8 

7.8 

7.6 

7.5 

7.4 

7.1 

6.7 

240 

7.8 

7.7 

7.7 

7.7 

7.7 

■  7.7 

7.8 

7.8 

7.7 

7.6 

7.6 

7.5 

7.2 

260 

7.8 

7.7 

7.7 

7.6 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.8 

7.8 

7.6 

280 

7.4 

7.4 

7.5 

7.5 

7.5 

7.5 

7.5 

7.5 

7.5 

7.6 

7.6 

7.8 

7.7 

300 

7.1 

7.2 

7.3 

7.3 

7.3 

7.3 

7.3 

7.4 

7.5 

74 

7.5 

7.5 

7.7 

320 

6.8 

6.9 

6.8 

7.0 

7.1 

7.1 

7.1 

7.1 

7.3 

7.3 

7.3 

7.4 

7.4 

340 

6.4 

6.5 

6.6 

6.6 

6.7 

6.7 

6.8 

6.9 

7.0 

7.1 

7.2 

7.2 

7.2 

360 

6.2 

6.2 

6.2 

6.3 

6.4 

6.4 

6.5 

6.6 

6.7 

6.7 

6.9 

6.9 

7.1 

380 

5.9 

5.8 

5.8 

5.9 

6.0 

6.1 

6.2 

6.3 

6.4 

6.4  6.4 

6.6 

6.8 

400 

5.6 

5.6 

5.6 

5.7 

5.7 

5.7 

5.8 

5.9 

5.9 

6.0  6.1 

6.2 

6.4 

420 

5.4 

5.4 

5.5 

5.5 

5.5 

5.5 

5.5 

5.5 

5.6 

5.6  5.6 

5.7 

5.8 

440 

5.6 

5.3 

5.3 

5.3 

5.3 

5.2 

5.2 

5.2 

5.2 

5.11  5.0 

5.3 

5.5 

460 

6.0 

5.6 

5.4 

5.3 

5.2 

5.2 

5.1 

5.0 

5.1 

5.2  5.2 

5.2 

5.3 

480 

6.5 

6.0 

5.7 

5.4 

5.2 

5.2 

5.1 

4.9 

4.9 

4.9  4.9 

5.0 

5.0 

500 

7.2 

6.8 

6.3 

5.9 

5.6 

5.3 

5.0 

4.8 

4.9 

4.8  4.8 

4.8 

4.9 

520 

8.0 

7.4 

7.0 

6.5 

6.1 

5.5 

5.4 

5.1 

4.9 

4.7  4.7 

4.7 

4.8 

540 

9.0 

8.4 

7.8 

7.3 

6.7 

63 

5.8 

5.4 

5.2 

4.9 

4.7 

4.7 

4.7 

560 

9.9 

95 

8.8 

8.2 

7.7 

7.1 

6.5 

6.0 

5.7 

5.3 

5.0 

4.8 

4.6 

580 

10.6 

10.2 

9.8 

9.3 

8.8 

8.1 

7.2 

6.8 

6.4 

6.0 

5.6 

5.1 

4.9 

600 

11.2 

11.0 

10.7 

10.3 

9.6 

9.1 

8.5 

7.7 

7.1 

6.4 

6.1 

5.6 

5.3 

620 

11.7 

11.5 

11.4 

11.0 

10.6 

9.9 

9.5 

8.9 

8.1 

7.4 

6.8 

6.3 

5.9 

640 

11.8 

11.9 

11.8 

11.7 

11.3 

11.0 

10.4 

9.8 

9.3 

8.5 

7.8 

7.1 

6.6 

666 

11.6 

11.8 

12.0 

12.1 

11.9 

11.6 

11.2 

10.8 

10.2 

9.6 

8.9 

8.2 

7.5 

680 

11.0 

11.6 

12.1 

12.2 

12.1 

12.2 

12.1 

11.5 

11.1 

10.6 

10.1 

9.2 

8.5 

700 

10.1 

10.9 

11.6 

12.1 

12-4 

12.3 

12.3 

12.3 

11.9 

11.4 

10.8 

10.4 

9.7 

720 

9.1 

10.0 

10.6 

11.4 

11.9 

12.4 

12.6 

12.5 

12.4 

12.0 

11.6 

11.2 

10.8 

740 

7.8 

8.8 

9.7 

10.5 

11.3 

11.8 

12.3 

12.8 

12.6 

12.6 

12.3 

11.9 

11.5 

760 

6.6 

7.6 

8.5 

9.4 

10.3 

11.0 

11.7 

12.1 

12.6 

12.8 

12.7 

12.5 

12.1 

780 

5.2 

6.3 

7.1 

8.1 

9.2 

10.1 

10.7 

11.6 

12.0 

12.4 

12.8 

12.9 

12.8 

800 

4.0 

4.8 

5.7 

6.7 

7.7 

8.7 

9.7 

10.5 

11. 3i 

11.9 

12.3 

12.5 

12.9 

820 

^M 

3.6 

4.4 

5.4 

6.4 

7.3 

8.4 

9.5 

10.3 

11.0 

11.7 

12.1 

12.5 

840 

2.2 

2.7 

3.3 

4.0 

4.9 

6.0 

7.0 

8.0 

9.1 

10.0 

10.8 

11.4 

12.0 

860 

1.6 

1.6 

2.2 

2.9 

3.6 

4.6 

5.6 

6.6 

7.6 

8.6 

9.6 

10.5 

11.2 

880 

1.2 

1.3 

1.5 

1.9 

2.6 

3.3 

4.1 

5.2 

6.1 

7.1 

8.2 

9.2 

10.1 

900 

l.l 

1.1 

1.2 

1.3 

1.7 

2.2 

2.9 

3.8 

4.8 

5.7 

6.8 

7.9 

8.8 

920 

1.1 

1.0 

1.0 

1.1 

1.1 

1.4 

1.9 

2.6 

3.4 

4.4 

5.3 

63 

7.4 

940 

1.5 

1.1 

0.8 

0.9 

1.0 

1.1 

1.3 

1.6 

2.3 

3.1 

3.9 

5.0 

5.9 

960 

23 

1.7 

1.3 

0.9 

0.7 

0.8 

0.9 

1.2 

1.4 

2.0 

2.8 

3.5 

4.6 

980 

30 

2.5 

1.9 

1.4 

1.2 

1.0 

0.8 

0.9 

1.2 

1.4 

1.7 

2.4 

3.3 

1000 

37 
800 

3.2 
810 

2.6 
820 

2.1 
830 

1.7 
840 

1.3 

0.9 

0.7 

0.7 

1.0 
890 

1.2 
900 

1.6 
910 

2.2 
920 

1 

850 

860 

870 

880 

88 


TABLE  XXXI. 


TABLE  XXXIL 


1 


Perturbations  by  Mars. 
Arguments  II.  and  IV. 
IV. 


Perfs.  by  Jupiter 
Arg's.  II.  and  V. 
V. 


n. 

0 

920 

930 
3.0 

940 

950 

960 
5.8 

970 
6.9 

980 
7.8 

990 

1000   0  1  10 

20 

30 

2.2 

3.8 

4.8 

8.4 

9.5 

15.3 

15.1 

15.0 

15.0 

20 

1.5 

2.1 

2.6 

3.4 

4.4 

5.5 

6.5 

7.6 

8.7 

14.9 

14.9 

14.7 

14.8 

40 

1.0 

1.4 

1.8 

2.5 

3.2 

4.0 

5.2 

6.0 

7.1 

14.7 

14.6 

14.6 

14.5 

60 

1.0 

1.1 

1.3 

1.8 

2.3 

3.0 

3.7 

4.8 

5.8 

14.4 

14.4 

14.4 

14.4 

80 

1.3 

1.1 

1.2 

1.4 

1.6 

2.2 

2.7 

3.6 

4.5 

13.4 

13.9 

14.0 

14.2 

100 

1.7 

1.3 

1.2 

1.2 

1.3 

1.6 

2.0 

2.6 

3.3 

13.2 

13.4 

13.6 

13.7 

120 

2.4 

2  0 

1.5 

1.4 

1.4 

1.4 

1.7 

1.9 

2.4 

12.3 

12.7 

13.0 

13.3 

140 

33 

2.8 

2.3 

2.0 

1.7 

1.5 

1.5 

1.8 

2.1 

11.3 

11.8 

12.1 

12.5 

160 

42 

3.6 

3.1 

2.6 

2.1 

2.0 

1.7 

1.7 

1.9 

10.2 

10.7 

11.2 

11.7 

180 

5.2 

4.6 

4.0 

3.5 

3.1 

2.5 

2.0 

2.0 

1.9 

9.1 

9.6 

10,1 

10.7 

200 

6.1 

5.5 

5.0 

4.4 

3.9 

3.5 

2.8 

2.7 

2.9 

7.8 

8.3 

8.9 

9.5 

220 

6.7 

6.3 

5.8 

5.4 

4.9 

44 

3.9 

3.2 

3.0' 

6.8 

7.2 

7.7 

8.3 

240 

7.2 

6.9 

6.6 

6.1 

5.6 

5  3 

4.8 

4.2 

3.7 

5.7 

6.2 

6.6 

7.2 

260 

7.6 

7.5 

7.1 

6.8 

6.5 

6.0 

5.6 

5.2 

4.8 

4.8 

5.2 

5.6 

6.1 

280 

7.7 

7.7 

7.5 

7.3 

7.1 

6.7 

6.3 

5.9 

5.5 

3.9 

4.1 

4.7 

5.2 

300 

7.7 

7.7 

7.7 

7.7 

7.4 

7.2 

7.0 

6.6 

6.1 

3.4 

3.5 

3.9 

4.3 

320 

7.4 

7.4 

7.6 

7.7 

7.6 

7  6 

7.3 

7.1 

6.9 

3.2 

3.1 

3.4 

3.6 

340 

7.2 

7.2 

7.3 

7.5 

7.7 

7.6 

7.6 

7.6 

7.7 

3.2 

3.0 

3.0 

3.1 

360 

7.1 

7.1 

7.1 

7.2 

7.2 

7.6 

7.6 

7.6 

7.5 

3.5 

3.2 

2.9 

2.9 

380 

6.8 

6.9 

7.0 

7.0 

7.0 

7.1 

7.3 

7.5 

7.5 

4.5 

4.0 

3.4 

3.1 

400 

6.4 

6.6 

6.6 

6.7 

6.7 

6.9 

7.0 

7.1 

7.3 

5.0 

4.3 

3.8 

3.5 

420 

5.8 

5.9 

6.2 

63 

6.6 

6.5 

6.7 

6.7 

6.9' 

6.1 

5.2 

4.6 

4.1 

440 

5.5 

5.6 

3.7 

5.8 

6.0 

6.1 

6.3 

6.5 

6.5 

7.5 

6.6 

5.8 

4.9 

460 

5.3 

5.3 

5.4 

5.7 

5.7 

5.7 

5.9 

6.1 

6.2 

9.0 

7.9 

7.0 

6.3 

480 

5.0 

5.0 

5.0 

5.1 

5.3 

5.4 

5.5 

5.6 

5.8 

10.5 

9.5 

8.5 

76 

500 

4.9 

4.9 

5.0 

5.0 

5.0 

5.1 

5.2 

5.3 

5.3 

12.3 

11.3 

10.0 

9.1 

520 

4.8 

4.8 

4.8] 

4.8 

4.8 

4.7 

4.9 

5.0 

5.1 

14.0 

12.7 

11.7 

10.7 

540 

4.7 

4.7 

4.6' 

4.6 

46 

4.5 

4.6 

4.6 

4.7 

15.6 

14.-) 

13.3 

12.3 

560 

4.6 

4.5 

4.5 

4.4 

4.5 

4.5 

4.5 

4.5 

44 

17.1 

16.1 

15.1 

14.0 

580 

4.9 

4.7 

4.6 

4.5 

4.4 

4.4 

4.4 

4.4 

4.2 

18.6 

17.4 

16.5 

15.7 

600 

5.3 

4.9 

4.8 

4.7 

4.5 

4.4 

4.4 

4.3 

4.1 

19.8 

19.0 

17.9 

17.0 

620 

5.9 

5.5 

5.1 

4.8 

4.6 

4.5 

4.4 

43 

4.2 

20.8 

20.1 

192 

18.4 

640 

6.6 

6.1 

5.6 

5.4 

5.0 

4.7 

46 

4.5 

43 

21.6 

20.9 

20.2 

19.5 

660 

7.5 

6.8 

6.3 

5.9 

5.5 

53 

4.9 

4.8 

4.6 

22.1 

21.6 

21.0 

20.4 

680 

8.5 

7.8 

7.3 

6.5 

6.1 

5.6 

54 

5.1 

4.8 

22.3 

22.0 

21.6 

21.2 

700 

9.7 

8.9 

8.1 

7.6 

7.0 

6.3 

5.9 

5.6 

5.3 

22.2 

22.0 

21.7 

21.5 

720 

10.8 

10.0 

9.3 

8.5 

7.9 

72 

6.0 

6.1 

5.8 

22.0 

21.9 

21.7 

21.6 

740 

11.5 

11.0 

10.2 

9.7 

8.9 

8.2 

7.6 

6.9 

6.5 

21.6 

21.6 

21.5 

21.5 

760 

12.1 

11.8 

11.3 

10.5 

10.0 

9.3 

8.5 

7.9 

7.3 

21.2 

21.1 

21.1 

21.2 

780 

12.8 

12.3 

11.9 

11.4 

10.9 

10.2 

9.6 

9.0 

8.2 

20.4 

20.5 

20.6 

20.7 

800 

12.9 

12.9 

13.5 

12.1 

11.7 

11.2 

10.5 

9.8 

9.2 

19.6 

19.8 

19.9 

20.1 

820 

12.5 

12.7 

12.8 

12.7 

12.2 

11.9 

11.2 

10.7 

10.1 

18.8 

190 

19.2 

19.4 

840 

12.0 

12.4 

12.6 

12.8 

12.6 

12  4 

12.2 

11.5 

10.9 

18.1 

18  2 

18.4 

18.6 

860 

11.2 

11.8 

12.3 

12.5 

12.7 

12  5 

12  5 

12.3 

11.7 

17.4 

17.5 

17.6 

17.9 

880 

10.1 

11.0 

11.5 

12.1 

12.3 

126 

12.6 

124 

12.3 

16.9 

16.9 

16.9 

17.1 

900 

8.8 

9.8 

10.6 

11.3 

11.8 

12.2 

t2.4 

12.5 

12.4 

16.3 

16.4 

16.4 

16.5 

920 

7.4 

8.4 

9.3 

10.2 

11.0 

11.5 

12.1 

12.2 

123 

16.0 

15.9 

15.9 

16.0 

940 

5.9 

7.1 

8.1 

8.9 

9.9 

10.7 

11.2 

11.7 

12.1 

15.8 

15.7 

15.7 

15.6 

9G0 

4.6 

5.6 

6.7 

7.7 

8.7 

9.4 

10.2 

10.9 

11.4 

15.5 

15.4 

15.3 

15.4 

980 

33 

4.2 

5.2 

6.2 

7.3 

8.2 

8.9 

9.9 

10.6 

15.3 

15.2 

15.2 

15.1 

1000 

2.2 

920 

3.0 
930 

3.8 

4.8 
9.50 

5.8 
960 

6.9 

7.8 

8.7 
990 

9.5 

15.3 

15.1 

15.0 

15.0 
30 

940 

970 

980 

1000 

1  0 

10 

20 

TABLE  XXXII. 


39 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


]„. 

30 

41) 

1  50 

60 
14.7 

70 

80 
14.5 

90 
14.5 

100 
14.4 

,  110 
14.5 

120 

1  130 

140 
14.7 

150 
14.8 

0 

15.0 

14.8 

14.7 

14.6 

14.5 

14.6 

20 

14.8 

14.7 

14.6 

14.4 

14.4 

14.2 

14.2 

14.1 

14.1 

14.1 

14.1 

14.1 

14.2 

40 

14.5 

14.4 

14.4 

14.3 

14.2 

14.1 

13.9 

13.8 

13.8 

13.8 

138 

138 

13.7 

60 

14.4 

14.3 

14.3 

14.2 

14.1 

13.9 

13.8 

13.6 

13.5 

13.5 

13.5 

13.4 

13.3 

80 

14.2 

14.2 

14.1 

14.5 

14.0 

13.8 

13.7 

13.5 

13.4 

13.2 

13.1 

13.0 

13.1 

100 

13.7 

13.7 

13.9 

13.9 

13.8 

13.7 

13.6 

13.5 

13.4 

13.2 

13.0 

12.8 

12.7 

120 

13.3 

13.4 

13.4 

1.3.5 

13.6 

13.5 

13.5 

13.3 

13.3 

13.2 

13.0 

12.8 

12.6 

140 

12.5 

12.8 

13.0 

13.1 

13.2 

13.2 

13.3 

13.2 

13.1 

13.0 

12.9 

12.8 

,126 

160 

11.7 

12.0 

12.4 

12.6 

12.7 

12.8 

12.9 

12.9 

13.0 

129 

12.8 

12.7 

I12.5 

180 

10.7 

11.1 

11.6 

11.9 

12.2 

12.3 

12.5 

12.5 

12.6 

12.7 

12.8 

126 

12.5 

200 

9.5 

10.0 

10.6 

11.0 

11.5 

11.7 

11.9 

12.2 

12.2 

12.3 

12.4 

12.3 

12.3 

220 

8.3 

8.8 

9.5 

9.9 

10.4 

10.8 

11.3 

11.5 

11.8 

11.9 

12.0 

12.0 

12.0 

240 

7.2 

7.7 

8.2 

8.9 

9.4 

9.8 

10.3 

10.6 

11.0 

11.3 

11.5 

11.7 

11.8 

260 

6.1 

6.5 

7.1 

7.6 

8.3 

8.8 

9.3 

9.7 

10.1 

10.5 

10.9 

II.O 

11.2 

280 

6.2 

5.6 

6.0 

6.5 

7.1 

7.6 

8.2 

8.7 

9.2 

9.6 

10.0 

10.4 

10.6 

300 

4.3 

4.7 

5.1 

5.5 

6.1 

6.6 

7.1 

7.6 

8.1 

8.7 

9.1 

9.4 

9.9 

320 

3.6 

3.9 

4.3 

4.6 

5.1 

5.4 

6.0 

6.6 

7.2 

7.7 

8.1 

8.5 

8.9 

340 

3.1 

3.3 

3.5 

3.8 

4.1 

4.5 

5.0 

5.4 

6.1 

6.6 

7.2 

7.6 

8.0 

360 

2.9 

3.0 

3.1 

3.3 

3.6 

3.8 

4.1 

4.5 

5.0 

5.5 

6.1 

6.6 

7.1 

380 

3.1 

2.8 

2.8 

2.7 

2.8 

2.9 

3.0 

3.2 

3.5 

4.1 

4.6 

5.0 

5.6 

400 

35 

3.1 

2.9 

29 

2.8 

2.8 

3.0 

3.1 

3.4 

3.8 

4.2 

'4.7 

5.2 

420 

4.1 

"3.6 

3.3 

3.1 

28 

27 

2.8 

2.9 

3.1 

3.2 

3.5 

3.8 

4.3 

440 

4.9 

4.4 

3.9 

34 

3.1 

27 

2.8 

27 

2.8 

3.1 

3.1 

3.2 

3.5 

460 

6.3 

5.4 

4.8 

4.3 

3.7 

3.2 

2.9 

2.8 

2.8 

2.7 

2.7 

2.8 

3.2 

480 

7.6 

6.7 

6.9 

5.2 

4.6 

4.1 

3.6 

3.1 

3.0 

2.8 

2.8 

2.6 

2.7 

500 

9.1 

8.1 

7.2 

6.4 

5.7 

5.0 

4.4 

3.9 

3.4 

3.2 

3.1 

2.9 

2.7 

520 

10.7 

9.5 

8.7 

7.7 

6.9 

6.1 

5.5 

4.8 

4.2 

3.8 

3.5 

3.2 

3.1 

540 

12.3 

11.1 

10.2 

9.1 

8.4 

7.4 

6.6 

5.9 

5.3 

4.7 

4.1 

3.8 

3.5 

560 

14.0 

13.0 

11.9 

10.8 

9.9 

8.7 

7.9 

7.1 

6.4 

5.8 

5.2 

4.5 

4.1 

580 

15.7 

14.5 

13.6 

12.5 

11.4 

10.4 

9.3 

8.3 

7.7 

6.9 

6.2 

5.5 

5.0 

600 

17.0 

16.0 

15.0 

14.0 

13.1 

120 

11.0 

10.1 

9.2 

8.2 

7.5 

6.7 

6.0 

620 

18.4 

17.4 

16.5 

15.5 

14.7 

13.6 

12.6 

11.6 

10.7 

9.8 

9.0 

8.0 

7.3 

640 

19.5 

18.5 

17.9 

17.0 

16.0 

15.1 

14.2 

13.1 

12.2 

11.3 

10.8 

9.4 

8.7 

660 

20.4 

19.7 

18.9 

18.1 

17.4 

16.3 

15.6 

14.6 

13.7 

12.8 

11.9 

11.0 

10.1 

680 

21.2 

20.5 

19.9 

19.1 

18.5 

17.6 

16.8 

16.0 

15.1 

14.2 

13.5 

12.5 

11.6 

700 

21.5 

21.0 

20.6 

20.0 

19.3 

18.7 

18.0 

17.1 

16.5 

15.6 

14.7 

13.8 

13.0 

720 

21.6 

21.2 

21.0 

20.5 

20.0 

19.3 

18.9 

18.3 

17.5 

16.8 

16.1 

1.0.1 

14.3 

740 

21.5 

21.2 

21.1 

20.8 

20.5 

20.0 

19.4 

18.9 

18.4 

17.7 

17.2 

16.8 

15.7 

760 

21.2 

21.0 

21.0 

20.8 

20.7 

20.3 

20.0 

19.4 

19.0 

18.6 

17.9 

17.4 

16.7 

780 

20.7 

20.7 

20.7 

20.6 

20.6 

20.3 

20.2 

19.8 

19.4 

19.1 

18.7 

18.1 

17.6 

800 

20.1 

20.2 

20.3 

20.3 

20.4 

20.3 

20.1 

19.9 

19.7 

19.3 

19.1 

18.7 

18.2 

820 

19.4 

19.5 

19.7 

19.8 

19.9 

19.9 

199 

19.8 

198 

19.6 

19.2: 

18.9 

18.7 

840 

18.6 

18.8 

18.9 

19.0 

19.2 

19.3 

19  4 

19.4 

19.4 

19.4 

19.4 

19.0 

18.9 

860 

17.9 

18.0 

18.3 

18.4 

18.6  1 

18.7 

18.8 

18.9 

19.0 

19.1 

19.1  ' 

19.0 

18.8 

880 

17.1 

17.2 

17.5 

17.6 

17.9 

18.0 

18.2 

18.3 

18.5 

18.6 

18.61 

18.6 

18.7 

900 

16.5 

16.6 

16.8 

16.9 

17.1 

17.1 

17.4 

17.5 

17.7 

17.9 

18.1 

18.2 

18.2 

920 

16.0 

16.0 

16.1 

16.2 

16.4 

16.5 

16.7 

16.8 

17.0 

17.2 

17.4 

17,5 

17.7 

940 

15.6 

15.5 

15.6 

15.6 

15.7  15.8 

16.0 

16.1 

16  3 

16.5 

16.8 

16.8 

17.1 

960 

15.4 

15.3 

15.3  15.2 

15.2  15.2 

15.3 

15.4 

15.6 

15.7 

15  9  1 

16.0 

16.3 

980 

15.1 

15.0 

15.0 

14.9 

14.9  14.8 

14.9 

14.9 

14.9 

15.0 

15.2 

15.3 

15.5 

1000 

15.0 

14.8 

14.7 

14.7 

14.6  14.5 

14.5 

14.4 

14.5 
110 

14.5 
120 

14.6 
130  1 

14.7 
140 

14.8 

30 

40 

50 

60 

70 

80  1 

90 

100 

150 

40 


TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 
Arguments  II.  and   V. 
V. 


II. 

0 

150 

160 

170 

180 
15.5 

190 

15.8 

200 

210 

220 
16.3 

230 

240 

250 

260 

270 

14.8 

15.0 

15.3 

15.9 

16.2 

16.7 

17.0 

17.1 

17.3 

17.6 

20 

14.2 

14.3 

14.6 

14.8 

14.9 

15.2 

15.5 

15.7 

15.9 

16.2 

16.6 

16.8 

17.1 

40 

13.7 

13.7 

13.9 

14.1 

14.3 

14.5 

14.8 

15.0 

15.3 

15.5 

15.8 

16.2 

16.4 

60 

13.3 

13.2 

13.4 

13.5 

13.6 

13.8 

14.1 

14.3 

14.6 

14.8 

15.1 

15.5 

15.8 

80 

13.1 

13.0 

13.0 

13.0 

13.1 

13.1 

13.3 

13.5 

13.8 

14.1 

14.4 

14.5 

15.1 

100 

12.7 

12.7 

12.7 

12.6 

12.7 

12.6 

12.8 

12.9 

13.1 

13.4 

13.7 

14.0 

14.2 

120 

12.6 

12.5 

125 

12.4 

12.3 

12.2 

12.3 

12.3 

12.6 

12.8 

130 

133 

13.6 

140 

12.6 

12.4 

12.4 

12.3 

12.1 

12.0 

12.0 

12.0 

12.1 

12.1 

12.3 

12.5 

12.8 

160 

12.5 

12.3 

12.2 

12.1 

12.1 

11.9 

11.8 

11.8 

11.8 

11.8 

11.9 

12.0 

12.2 

180 

12.5 

12.3 

12.2 

12.1 

11.9 

11.8 

11.7 

11.5 

11.5 

11.5 

11.6 

11.7 

11.8 

200 

12.3 

12.2 

12.2 

12.0 

11.9 

11.7 

11.7 

11.5 

11.4 

11.3 

11.2 

11.3 

H.5 

220 

12.0 

12.0 

12.1 

12.0 

11.8 

11.6 

11.6 

11.5 

11.4 

11.3 

11.2 

11.1 

11.1 

240 

11.8 

11.8 

11.9 

11.9 

11.8 

11.6 

11.5 

11.4 

11.3 

11.2 

11.1 

11.1 

11.0 

260 

11.2 

11.5 

11.6 

11.6 

11.6 

11.5 

11.3 

11.3 

11.3 

11.2 

11.1 

11.0 

10.9 

280 

10.6 

10.8 

11.1 

11.2 

11.2 

11.2 

11.3 

11.3 

11.2 

11.2 

11.1 

11.0 

10.9 

300 

9.9 

10.1 

10.5 

10.8 

10.9 

11.0 

11.1 

11.0 

11.0 

11.0 

11.0 

11.1 

10.9 

320 

8.9 

9.4 

9.7 

10.1 

10.4 

10.5 

10.7 

10.8 

10.8 

10.8 

10.8 

10.8 

10.9 

340 

8.0 

8.5 

9.1 

9.3 

9.6 

9.9 

10.2 

10.3 

10.5 

10.6 

10.6 

10.7 

10.7 

360 

7.1 

7.5 

8.0 

8.4 

8.9 

9.2 

9.5 

9.8 

10.1 

10.3 

10.4 

10.5 

10.5 

380 

5.6 

6.2 

6.8 

7.3 

7.8 

8.3 

8.9 

9.3 

9.7 

10.0 

10.0 

10.1 

10.2 

400 

5.2 

5.6 

6.2 

6.6 

7.0 

7.5 

7.9 

8.4 

8.8 

9.1 

9.4 

9.7 

9.9 

420 

4.3 

4.8 

5.3 

5.8 

6.2 

6.6 

7.1 

7.4 

7.9 

8.4 

8.7 

9.1 

9.4 

440 

3.5 

3.9 

4.4 

4.9 

5.4 

5.7 

6.2 

6.7 

7.1 

7.6 

7.9 

8.4 

8.7 

460 

3.2 

3.3 

3.8 

4.1 

4.5 

4.9 

5.4 

5.7 

6.3 

6.7 

7.2 

7.7 

8.0 

480 

2.7 

2.9 

3.2 

36 

3.9 

4.3 

4.7 

5.0 

5.4 

•5.9 

6.3 

6.8 

7.3 

500 

2.7 

2.7 

2.9 

3.1 

3.4 

3.6 

4.0 

4.4 

4.8 

5.2 

5.7 

5.9 

6.4 

520 

3.1 

2.8 

2.9 

3.0 

3.1 

3.2 

3.5 

3.8 

4.2 

4.7 

4.9 

5.4 

5.7 

540 

3.5 

3.2 

3.1 

3.0 

3.0 

3.0 

3.3 

3.5 

3.7 

4.1 

4.3 

4.7 

5.1 

560 

4.1 

3.8 

3.6 

3.3 

3.2 

3.2 

3.2 

3.3 

3.5 

3.7 

4.0 

4.3 

4.5 

580 

5.0 

4.6 

4.2 

4.0 

3.6 

3.5 

3.3 

3.2 

3.4 

3.5 

3.7 

4.0 

4.2 

600 

6.0 

5.4 

5.1 

4.6 

4.3 

3.9 

3.7 

3.5 

3.5 

3.6 

3.7 

3.8 

4.0 

620 

7.3 

6.6 

6.0 

5.6 

5.1 

4.6 

4.2 

4.0 

3.9 

3.8 

39 

3.9 

4.0 

640 

8.7 

7.8 

7.3 

6.6 

6.1 

5.5 

5.2 

4.7 

4.4 

4.2 

4.0 

4.0 

4.1 

660 

10.1 

9.3 

8.6 

7.7 

7.2 

6.5 

6.2 

5.9 

53 

4.9 

4.6 

4.5 

4.4 

680 

11.6 

10.8 

100 

9.3 

8.5 

7.5 

7.3 

6.7 

6.3 

5.8 

5.5 

5.2 

4.9 

700 

13.0 

12.1 

11.5 

10.7 

9.9 

9.0 

8.5 

7.8 

7.4 

6.9 

6.3 

6.0 

5.8 

720 

14.3 

13.5 

12.8 

12.1 

11.3 

10.6 

9.8 

9.1 

8.7 

8.0 

7.6 

7.0 

6.6 

740 

15.7 

14.9 

14.2 

13.4 

12.7 

120 

11.2 

10.5 

9.7 

9.3 

8.9 

8.2 

7.7 

760 

16.7 

15.9 

15.5 

14.7 

13.9 

13.3 

12.6 

11.8 

11.2 

10.5 

10.0 

9.5 

9.0 

780 

17.6 

17.0 

16.4 

15.7 

15.1 

14.6 

13.8 

13.2 

12.6 

11.9 

11.2 

10.8 

10.2 

800 

18.2 

17.8 

17.3 

16.8 

16.2 

16.0 

15.0 

14.3 

13.7 

13.1 

12.6 

12.0 

11.5 

820 

18.7 

18.3 

18.0 

17.6 

17.0 

16.6 

16.0 

15.3 

14.9 

14.3 

13.7 

13.1 

12.6 

840 

18.9 

18.7 

18.4 

18.2 

17.7 

17.2 

16.8 

16.3 

15.8 

15.3 

14.9 

14.4 

13.8 

860 

18.8 

18.7 

18.6 

18.4 

18.3 

17.9 

17.4 

17.1 

16.7 

16.3 

15.9 

15  4 

15.0 

880 

18.7 

18.5 

186 

18.5 

183 

18.2 

18.0 

17.7 

17.4 

17.1 

16.6 

16.3 

15.9 

900 

18.2 

18.2 

18.3 

18.3 

18.3 

18.1 

18.1 

18.0 

17.8 

17.6 

173 

17.0 

16.7 

920 

17.7 

17.9 

18.0 

18.0 

18.1 

18.1 

18.0 

18.0 

18.0 

17.8 

17.7 

17.6 

17.3 

940 

17.1 

17.1 

17.4 

17.6 

17.6 

17.7 

17.8 

17,8 

17.9 

18.0 

17.8 

17.8 

17.7 

960 

16.3 

16.5 

16.8 

169 

17.1 

17.2 

17.4 

17.5 

17.6 

17.8 

17.9 

18.0 

179 

980 

15.5 

15.7 

16.1 

16. 3 

16.5 

16.7 

168 

17.0 

17.2 

17.3 

17.6 

17.7 

17.9 

1000 

14.8 

15.0 

15.3 

15.5 

15.8 

15.9 

16.2 
210 

16.3 

16.7 

17.0 

17.1 

17.3 

17.5 

150 

160 

170 

180 

190 

200 

220 

230 

240 

250 

260 

270 

J 

TABLE  XXXII. 


41 


Perturbations  jvuduced  by  Jupiter, 
Arguments  II.  and  V 
V 


II. 

0 

270  1  230  '  290 

'          i 

1  300 

1  310 

320 

330 

340 

350 

360  370 

380 

390 

17.5 

17.5 

17.7 

17.8 

17.9 

17.9 

18.0 

18.0 

17.9 

17.7 

17.6 

17.5 

17.5 

20 

17.1 

17.3 

17.5 

17.6 

17.8 

17,8 

18.0 

18.1 

18.1 

18.1 

18,0 

18.0 

18.0 

40  16.4 

16.8 

16.9 

17.2 

17.6 

17.7 

17.9 

18.1 

18.3 

18.3  18.4 

18.4 

18.6 

60  15.8 

16.0 

16.4 

16.7 

10.9 

17.3 

17.6 

17.9 

18.2 

18.3  I  18.5 

18.5 

18.7 

80 

15.1 

15.4 

15.7 

16.1 

16.4 

16.7 

17.0 

17.5 

17.8 

18.0 

18.3 

18.5 

18.8 

100 

14.2 

14.6 

15,1 

15.0 

15.8 

16.1 

10.5 

17.0 

17.2 

17.5 

17.9 

18.3 

18.7 

120 

13.6 

137 

14.2 

14.5 

15.0 

15.4 

15.8 

16.2 

16.7 

17.1 

17.3 

17.9 

18.3 

140 

12.8 

13.1 

13.3 

13.7 

14.2 

14.4 

15.1 

15.5 

15.9 

16.3 

16.8 

17.3 

17.7 

160 

12.2 

12.4 

12.6 

12.9 

13.4 

13.8 

14.1 

14.6 

15.2 

15.5 

16.0 

16.5 

17.1 

180 

11.8 

11.9 

13.1 

12.3 

12.5 

12.8 

13.3 

13.7 

14.4 

14.7 

15.2 

15.7 

16.3 

200 

11.5 

11.5 

11.6 

11.7 

12.0 

12.1 

12.5 

13.0 

13.4 

13.8 

14.3 

14.7 

15.5  1 

220 

11.1 

11.1 

11.2 

11.3 

11.6 

11.7 

11.9 

12.3 

12.7 

13.0 

13.5 

14.0 

14.5 

240 

11.0 

10.9 

10.9 

11.0 

11.2 

11.3 

11.5 

11.8 

12.1 

12.3 

12.8 

13.2 

13.8 

260 

10.9 

10.8 

10.8 

10.8 

10.9 

10.9 

11.1 

11.3 

11.4 

11.6 

12.0 

12.3 

13.0 

280 

10.9 

10.8 

10.7 

10.6 

10.7 

10.6 

10.8 

11.0 

11.2 

11.3 

11.5 

11.8 

12.2 

300 

10.9 

10.8 

10.7 

10.6 

10.6 

10.5 

10.6 

10.7 

10.8 

10.9 

11.1 

11.4 

11.8 

320 

109 

10.7 

10.7 

10.6 

10.6 

10.5 

10.5 

10.6 

10.7 

10.6 

10.7 

11.0 

11.2 

340 

10.7 

10.7 

10.6 

10.5 

10.5 

10.4 

10.5 

10.5 

10.6 

10.5 

10.6 

10.7 

10.8 

360 

10.5 

10.5 

10.5 

10.5 

10.5 

10.4 

10.4 

10.4 

10.4 

10.3 

10.5 

10.6 

10.8 

380 

10.2 

10.3 

10  3 

10.3 

10.4 

103 

10.4 

104 

10.4 

10.3 

10.3 

10.4 

10.6 

400 

9.9 

10.0 

10.0 

10.2 

10.3 

10.2 

10.2 

103 

10.4 

10.3 

10.3 

10.3 

10.5 

420 

9.4 

96 

9.8 

9.9 

10.1 

102 

10.1 

10.2 

10.2 

10.2 

10.3 

10.3 

10.4 

440 

8.7 

9.0 

9.2 

9.4 

9.7 

9.8 

10.0 

10.1 

10.2 

10.1 

10.1 

10.2 

10.4 

400 

80 

84 

8.6 

8.8 

9.1 

9.3 

9.6 

9.9 

10.1 

10.0 

10.0 

10.2 

10.3 

480 

73 

7.6 

79 

8.4 

8.7 

8.9 

9.1 

9.4 

9.6 

9.7 

9.8 

10.0 

10.1 

500 

6.4 

6.9 

7.2 

7.6 

8.0 

8.3 

8.6 

8.9 

9.2 

9.4 

9.5 

9.7 

9.9 

520 

5.7 

n.i 

66 

6.9 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

9.1 

9.4 

9.7 

540 

5.1 

5.4 

5.8 

6.2 

6.7 

7.0 

7.4 

7.7 

8.0 

•  8.3 

8.6 

8.9 

9.2 

560 

4.5 

4  9 

5.1 

5.5 

6.0 

6.3 

67 

7.2 

7.5 

7.7 

8.0 

8.3 

8.7 

580 

4.2 

4.4 

4.8 

5.0 

5.3 

5.7 

6.1 

6.6 

6.9 

7.1 

7,4 

7.7 

8.1 

OOJ 

40 

4.2 

4.3 

4.7 

4.9 

5,2 

5.6 

6.0 

6.3 

6.5 

6.8 

7.2 

7.6 

020 

4.0 

40 

4.1 

4.3 

4.7 

4.8 

5.1 

5.5 

5.8 

6  1 

0.4 

6.7 

7.0 

640 

4.1 

4.1 

4.2 

4.2 

4.4 

4.6 

4.8 

5.1 

5.4 

5.6 

5,9 

6  3 

6.6 

660 

44  1  43 

43 

43 

4.5 

4.5 

4.7 

4.9 

5.1 

5.3 

5,5 

5.8 

6.2 

6S0 

4.9 

4.9 

4.7 

4.6 

4.7 

4.5 

46 

4.8 

5.0 

5.1 

53 

5.5 

5.8 

700 

5.8 

5.4 

5.2 

5.1 

5.0 

4.9 

4.9 

4.9 

5.1 

5.2 

5,3 

5.4 

5.6 

720 

6.6 

6.2 

5.9 

5.7 

5.6 

55 

5.4 

5.3 

5.3 

5.3 

5,3 

5.4 

5.5 

740 

7.7 

7.2 

68 

6.5 

6.4 

6.1 

6.0 

5.9 

5.8 

5.7 

5  6 

5.5 

5.7 

760 

9.0 

8.2 

79 

7.5 

7.2 

6.9 

6.7 

6.5 

6.3 

6.1 

5.9 

5.9 

6.0 

780 

10.2 

9.7 

9.1 

8.4 

8.2 

7.7 

7.6 

7.4 

7.2 

6.9 

6.6 

6.5 

65 

800 

11.5 

11.0 

10.4 

9.8 

9.4 

8.7 

8.5 

8.3 

8.0 

7.7 

7.6 

7.3 

7.1 

820 

12.6 

12.1 

11.7 

11.2 

106 

10.1 

9.7 

9.2 

9.1 

8.6 

8,3 

8.1 

7.9 

841 

138 

13.2 

12.8 

12.3 

11.9 

11.3 

10.9 

10.5 

10.2 

9.6 

9.4 

9.1 

8.9 

860 

15.0 

14.4 

13.8 

13.5 

13.1 

12.6 

12.1 

11.7 

11.2 

10.7 

10.4 

10.1 

10.0 

880 

15.9 

15.4 

150 

14.4 

14.2 

13.7 

13.4 

12  9 

12.5 

12.0 

11.5 

113 

11.1 

900 

16.7 

16.4 

15.9 

15.5 

15.2 

14.8 

14.4 

14.1 

13.7 

13.2 

12.8 

12.4 

12.2 

920 

17.3 

17.1 

168 

16.5 

16.2 

15,7 

15.5 

15.2 

14.8 

14  3 

14.0 

13.6 

13.3 

940 

17.7 

17.5 

17.3 

17.1 

169 

16.6 

16.3 

16.1 

16.0 

15.5 

15.0 

14.7 

14.5 

960 

17.9 

17.8 

17.6 

17.5 

17.4 

17.2 

17.0 

16.9 

16.8 

16.4 

16.2 

15  8 

156 

P80 

17.9 

17.8 

17.8 

17.8 

17.8 

17.8 

17.6 

17.5 

17.3 

17.2 

17.0 

16.8 

16.6 

1000 

17.5 
270  i 

17.7 
280 

17.7 

17.8 
300 

17.9 

17.9 

18.0 

18.0 
340 

17.9 
350 

17.7 
360 

17.6 
370 

17.5 
380 

17.5 

290 

310 

320  1 

330 

390 

42 


TABLE  XXXII. 


Perturbations  produced  by  Jupiter, 

Arguments  II.  and  V. 

V. 


II. 

0 

390 
17.5 

400 

17.1 

410 

420 
16.7 

430 

440 

450 
16.1 

460 
15.8 

470 

480 
15.1 

490 

."iOO 
14.3 

51l> 

17.0 

16.5 

16.3 

15.6 

14.6 

13.9 

20 

18.0 

18,1 

17.7 

17.5 

17.5 

17.2 

17.1 

16.8 

16.7 

16.3 

16.0 

15.0 

1&.3 

40 

18.6 

18.6 

18.5 

18.4 

18.3 

18.1 

180 

17.8 

17.6 

17.3 

17.2 

16.8 

16.5 

60 

18.7 

18.9 

18.9 

18.9 

18.9 

18.7 

18.8 

18.6 

18.7 

18.4 

18.1 

17.9 

17.7 

80 

18.8 

18.9 

19.2 

19.3 

19.4 

19.3 

19.3 

19.3 

19.3 

19.2 

19.2 

18.9 

18.8 

100 

18.7 

18.9 

19.1 

19.4 

19.7 

19.8 

19.8 

19.8 

19.8 

19.8 

19.9 

19.7 

19.7 

120 

18.3 

18.6 

18.9 

19.2 

19.5 

19.8 

20.0 

20.1 

20.3 

20.3 

20.4 

204 

20.4 

140 

17.7 

18.2 

18.6 

18.9 

19.2 

19.6 

20.0 

20.3 

205 

20.6 

20.7 

20.8 

21.0 

160 

17.1 

17.0 

17.9 

18.5 

19.0 

19.3 

19.8 

20.2 

20.5 

20.6 

20.9 

21.1 

21.2 

180 

10.3 

16.8 

17.3 

17.9 

18.3 

18.8 

19.3 

19.8 

203 

20.6 

20.9 

21.1 

21.4 

200 

15.5 

10.0 

16.5 

17.1 

17.7 

18.2 

18.6 

19.1 

19.8 

20.2 

20.7 

21.0 

21.4 

220 

14.5 

15.0  15.6 

16.1 

16.9 

17.4 

18.0 

18.6 

19.0 

19.7 

20.3 

20.7 

21.1 

240 

13.8 

14.2  14.7 

15.2 

15.9 

16.5 

17.1 

17.7 

18.4 

18.9 

19.5 

20.1 

20.7 

260 

13.0 

13.4 

13.9 

14.4 

15.0 

15.5 

16.3 

16.9 

17.5 

18.0 

18.6 

193 

20.0 

280 

12.2 

12.7 

13.0 

13.5 

14.2 

14.7 

15.3 

15.9 

16.7 

17.2 

17.8 

184 

19.1 

300 

11.8 

11.9 

12.4 

12  8 

13.3 

13.8 

14.4 

14.9 

15.7 

16.3 

17.0 

17.6 

18.2 

320 

11.2 

11.5 

11.8 

12.2 

12.7 

13.0 

13.6 

14.1 

14.7 

15.3 

16.0 

166 

17.4 

340 

10.8 

11.2 

11.4 

11.6 

12.1 

12.4 

12.9 

13  4 

13.9 

14.4 

15.1 

15  7 

16  4 

360 

10.8 

10.8 

11.0 

11.2 

11.6 

11.9 

12.3 

12.6 

1.3.2 

13.6 

14.2 

14.8 

1.5.5 

380 

10.6 

10.6 

10.7 

10.9 

11.2 

11.4 

11.9 

12.2 

12.6 

12.9 

13.5 

139 

14.5 

400 

10.5 

10.5 

10.6  10.6 

10.9 

11.1 

11.4 

11.8 

12.2 

12.5 

12.9 

13.3 

13.8 

420 

10.4 

10.4 

10.5 

10.6 

10.7 

10.9 

11.2 

11.3 

11.7 

11.9 

12.4 

12.8 

13.3 

440 

10.4 

10.4 

10.4 

10.5 

10.7 

10.8 

10.9 

11.1 

11.3 

11.6 

11.9 

122 

12.7 

460 

10.3 

10.4 

10.4 

10.4 

10.6 

10.6 

10.7 

10.9 

11.2 

11.3 

11.7 

11.9 

122 

480 

10.1 

10.2 

10.3 

10.4 

10.6 

10.6 

10.7 

10.8 

11.0 

11.2 

11.4 

11.7 

120 

500 

9.9 

10.0 

10.1 

10.2 

10.4 

10.5 

10.7 

10.8 

10.9 

11.0 

11.2 

11.3 

11.7 

520 

9.7 

9.8 

9.8 

10.0 

10.2 

10.3 

10.5 

10.6 

10.9 

10.8 

11.1 

11.3 

11.5 

540 

9.2 

9.4 

9.6 

9.8 

10.0 

10.2 

10.3 

10.4 

10.6 

10.7 

10.9 

11.1 

11.4 

560 

8.7 

89 

9.1 

9.3 

9.7 

9.8 

10.1 

10.3 

10.5 

10.6 

10.7 

108 

11.2 

580 

8.1 

8.5 

8.7 

8.7 

9.2 

9.4 

9.7 

9.9 

10.2 

10.4 

10.6 

10.7 

10.9 

600 

7.6 

7.9 

8.2 

8.5 

8.8 

9.0 

9.3 

9.5 

9.8 

10.0 

10.3 

10.5 

10.7 

620 

7.0 

7.3 

7.6 

7.9 

8.2 

8.5 

8.8 

90 

9.4 

9.6 

10.0 

10.1 

10.4 

040 

66 

6.8 

7.1 

7.4 

7.7 

7.9 

8.2 

8.6 

8.9 

9.1 

9.4 

9.7 

10.1 

660 

62 

6.4 

6.6 

6.9 

73 

7.6 

7.9 

8.1 

8.3 

8.6 

8.9 

9.2 

9.5 

080 

5.8 

6.1 

6.2 

6.5 

6.8 

7.0 

7.4 

7.6 

7.9 

8.1 

8.4 

8.7 

9.0 

700 

5.6 

5.8 

6.0 

6.2 

6.4 

6.6 

6.9 

7.1 

7.4 

7.6 

7.9 

8.2 

8.5 

720 

5.5 

5.6 

5.7 

5.9 

62 

6.3 

6.5 

6.8 

7.1 

7.2 

7.5 

7.7 

8.0 

740 

57 

5.7 

5.7 

5.8 

6.0 

6.1 

6.2 

6.4 

6.7 

6.9 

7.1 

7.2 

7.5 

760 

60 

6.0 

6.0 

6.0 

6.0 

6.1 

6.2 

63 

6.4 

6.5 

6.7 

6.8 

7.1 

780 

6.5 

6.3 

6.2 

6.2 

6.3 

6.3 

6.3 

6.3 

6.4 

6.4 

6.5 

6.7 

6.8 

800 

7.1 

7.0 

6.7 

6.6 

6.7 

6.5 

6.5 

6.4 

6.5 

6.5 

6.5 

6.6 

a.  7 

820 

7.9 

7.6 

7.5 

7.3 

7.2 

7.0 

7.0 

6.8 

6.8 

6.7 

6.6 

6.6 

6.7 

840 

8.9 

8.6 

83 

8.1 

7.8 

7.7 

7.6 

7.4 

7.3 

7.1 

7.0 

6.8 

6.8 

860 

10.0 

9.7 

9.3 

90 

8.7 

8.4 

8.2 

8.1 

7.9 

7.7 

7.6 

7.3 

7.2 

880 

11.1 

10.5 

10.4 

10.0 

9.7 

9.5 

9.2 

8.9 

8.7 

8.4 

8.2 

7.9 

7.7 

900 

12.2 

11.8 

11.5 

11.0 

10.8 

10.5 

10.3 

9.9 

9.7 

9.4 

9.0 

8.8 

8.5 

920 

133 

13.0 

12.6 

123 

12.1 

11.5 

11.3 

11.0 

10.6 

10.2 

10.1 

9.7 

9.4 

940 

14.5 

14.1 

13.8 

13.5 

13.2 

12.8 

12.5 

11.9 

11.8 

11.3 

11.0 

10.7 

10.4 

960 

15.6 

15.3 

14.9 

14.6 

14.4 

14.0 

13.7 

13.3 

130 

12.5 

12.1 

11.8 

11.5 

980 

16.6 

16.3 

160 

15.7 

15.6 

152 

14.9 

146 

14.2 

13.8 

136 

12.9 

12.7 

1000 

17.5 

17.1 

17.0 

16.7 

16.5 

16.3 

16.1 

15.8 

15.6 

15.1 

14.6 

14.3 

13.9 

390 

400 

410 

420 

430 

440 

450 

460 

470 

480 

490 

500 

510 

TABLE  XXXII. 


43 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


11. 

510 

520 

530 

540 

550 

560 

570 

580 
10.8 

590 

600 

610 
9.4 

620 
8.9 

630 

8.4 

0 

"  1  " 
13.9  13.4 

13.1 

12.7 

12.1  11.8  11.3 

10.2 

9.9 

20 

15.3  14.9 

14.4 

139 

13.5  13.1  12.5 

12.1 

11.5 

11.0 

10.4 

10.0 

9.4 

40 

16.5  16.3  15.7 

15  4 

15.0  14.3  13.8 

13.4 

12.8 

123 

11.7 

11.1 

10.5 

GO 

17.7  173 

17.0 

16.6 

16.1  15.8  15.3 

14.7 

14.3 

13.7 

13.0 

12.4 

11.8 

80 

18.8  18.5 

18.1 

17.9 

17.4  17.1  16.6 

16.2 

15.7 

15.1 

14.5 

13  9 

13.2 

100 

19.7 

19.5 

19.2 

19.0 

18.8  18.4  17.9 

17.6 

17.0 

16.5 

16.0 

15.2 

14.7 

120 

20.4 

203 

20.2 

200 

19.7  195  19.1 

18.8 

18.4 

18.0 

17.3 

16.8 

162 

140 

21.0 

21.1 

21.0 

20.8 

20.7  20.4  20.2 

19.9 

19.0 

19.3 

18  8 

18  3 

17.7 

160 

21.2 

21.5 

21.5 

21.6 

21.5  21.3  21.2 

21.0 

206 

20.4 

20.1 

19.6 

19.1 

180 

21.4 

21.6 

21.8 

22.0 

22.0  22.1  21.9 

21.8 

21. G 

21.4 

21.1 

20.7 

20  3 

200 

21.4 

21.7 

21.9 

22.1 

22.3  1  22.5  ,  22.5 

22.5 

22.4 

22.3 

22.1 

21.8 

21.5 

220 

21.1 

21.5 

21.8 

22.2 

22.5 

22.8 

23.1 

23.1 

22.9 

228 

22.9 

22.G 

225 

240 

20.7 

21.1 

21.5 

21.9 

22.3 

22  7 

23.0 

23.3 

23.4 

23.5 

23.4 

23  3 

23.2 

260 

20.0 

20.6 

21.0 

21.6 

22.0 

22.4 

22.8 

23.2 

23.5 

23.8 

23.8 

23.8 

23  9 

280 

19.1 

19.9 

20.4 

20  9 

21.5 

22.0 

22.4 

23.0 

23  3 

23.7 

240 

24.1 

24.1 

300 

18.2 

19.0 

19.6 

20.3 

20.7 

21.3 

21.8 

22.3 

23.0 

23.4 

23.8 

24.1 

24.3 

.320 

17.4 

18.9 

18.7 

19.4 

200 

20.6 

21.1 

21.8 

22  3 

22.9 

23.3 

23.7 

24.2 

340 

164 

170 

17.6 

18.5 

19.2 

19.9 

20.4 

21.1 

21.6 

22.2 

22.8 

23.3 

23.7 

360 

15.5 

16  2 

16.7 

17.4 

18.2 

18.9 

19.5 

20.1 

208 

21.5 

22.0 

22.6 

23.2 

380 

145 

15  2 

15.9 

16.6 

17.1 

17.9 

18.6 

19.3 

19.8 

20.5 

21.1 

21.8 

22.5 

400 

13.8 

14.4 

14.9 

15.6 

16.2 

16.8 

17.6 

18.4 

19.1 

19.7 

20.3 

20.9 

21.5 

420 

133 

13.7 

14.2 

14.8 

15.3 

16.0 

16.5 

17.4 

18.0 

18.7 

19.4 

20.0 

20.6 

440 

12.7 

13  1 

13.6 

14.1 

14.6 

15.2 

15.7 

16.4 

17.1 

17.8 

18.4 

18.9 

19.6 

460 

12.2 

12.7 

13  0 

13.5 

13.9 

14.4 

15.0 

15.6 

16.1 

16.9 

17.5 

18.2 

18.7 

480 

120 

12.2 

12.5 

130 

13.4 

13.9 

14.3 

14.8 

15.3 

15.9 

16.6 

17.3 

17.9 

500 

11.7 

12.0 

12.2 

12.6 

12.9 

13.3 

13.8 

14.3 

14.7 

15.2 

15.7 

16.4 

16.9 

.520  11.5 

11.9 

12.0 

12.3 

12.6 

13.0 

13.2 

13.8 

14.2 

14.7 

15.1 

15.5 

16.2 

.540  11.4 

11.6 

11.9 

12.2 

12.4 

12.7 

12.9 

13.3 

13.7 

14.2 

14.6 

15.0 

15  4 

.560  11.2 

11.4 

11.5 

11.9 

12.1 

12.4 

12.7 

13.1 

13.4 

13.8 

14.1 

14.5 

14.9 

.580  10.0 

11.2 

11.4 

11.6 

11.9 

12.2 

12.4 

12.8 

13.1 

13.5 

13.8 

14.2 

14.5 

600  10.7 

10.8 

11.1 

11.5 

11.7 

12.0 

12.2 

12.5 

12.8 

13.1 

13.4 

13.8 

14.2 

620  10.4 

10.7 

10.7 

11.1 

11.4 

11.6 

12.0 

12.3 

12.5 

12.9 

13.1 

13.4 

13.8 

640  10.1 

10.4 

10.6 

10.7 

11.0 

11.3 

11.6 

12.0 

12.3 

12.6 

12.9 

13.2 

13.5 

660   9.5 

9.9 

102 

105 

10.6 

11.0 

11.3 

11.6 

11.9 

12.3 

126 

12.9 

13.2 

680 

9.0 

9.3 

9.6 

10.0 

10.3 

10.5 

10.8 

11.3 

11.5 

11.9 

12.2 

12.4 

12.8 

700 

8.5 

8.9 

9.1 

9.5 

9.8 

10.1 

10.3 

10.7 

11.1 

11.4 

11.8 

12.1 

12.4 

720 

S.O 

8.3 

8.5 

9.0 

9.2 

9.6 

9.9 

10.2 

10.5 

10.9 

11.3 

11.7 

12.0 

740 

7.5 

7.8 

8.0 

8.3 

8.6 

9.0 

9.3 

9.7 

9.9 

10.4 

10.8 

11.1 

11.5 

760 

7.1 

73 

7.5 

79 

8.1 

8.4 

8.6 

9.1 

9.4 

9.7 

10.1 

10.5 

10.9 

780 

6  8 

7.0 

7.1 

7.3 

7.6 

7.9 

8.1 

8.5 

8.8 

9.2 

9.4 

9.8 

10.2 

800 

€.7 

6.8 

6.8 

7.0 

7.1 

7.3 

7.5 

7.8 

8.2 

8.5 

8.8 

9.1 

9.5 

820 

6.7 

6.8 

6.6 

6.8 

6.9 

7.0 

7.1 

7.4 

7.6 

7.9 

8.1 

8.4 

8.7 

840 

6.8 

6.8 

6.8 

6.8 

6.8 

6.9 

6.9 

7.1 

7.2 

7.4 

7.6 

7.9 

8.1 

800 

7.2 

7.1 

7.1 

7.0 

6.9 

69 

6.8 

6.8 

6.9 

7.1 

7.2 

73 

7.6 

880 

7.7 

7.5 

7.4 

7.3 

7.1 

7.0 

6.8 

6.8 

6.7 

6.8 

6.8 

7.0 

7.2 

900 

8.5 

8.2 

7.9 

7.7 

7.5 

7.3 

7.2 

7.1 

6.9 

6.9 

6.8 

6.8 

6.8 

920 

9.4 

92 

8.7 

8.4 

8.1 

7.9 

7.6 

7.4 

7.1 

7.0 

6.9 

6.8 

6.7 

940 

10  4 

10.0 

9.7 

9.4 

8.9 

8.6 

8.3 

8.1 

7.7 

7.4 

7.1 

6.9 

6.7 

960 

115 

112 

10.7 

10  4 

9.8 

9.5 

9.1 

88 

8.5 

8.1 

7.7 

7.4 

7.1 

980 

12  7 

12.3 

11.8 

11.5 

11.1 

10.6 

100 

9.7 

9.2 

8.9 

8.5 

8.1 

7.7 

1000 

139 
510 

13.4 
520 

13.1 

12.7 

12.1 
550 

11.8 

11.3 

10.8 
580 

10.2 

9.9 
600 

9.4 
610 

8.9 

62  ti 

8.4 

530 

540 

560 

570 

590 

630 

44 


TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  11.  and  V. 

V. 


II. 

630 

8.4 

640 

650 

660  '  670 

680  1 

690 

700 

710 

720  730 

740  j 

750 

0 

8.0 

7.7 

■« 

6.9 

6.7 

6.5 

6.5 

6.3 

6.2  6.2 

6.4 

6.5 

20 

9.4 

9,0 

8.4 

8.0 

7.5 

7.1 

6.9 

6.7 

6.4 

6.3  6.0 

6.1 

0.1 

40 

10.5 

10.1 

9.4 

8.9 

8.3 

7.8 

7.4 

7.0 

6.6 

6.4  6.2 

59 

5.8 

60 

11.8 

11.3 

10.6 

10.1 

9.3 

8.7 

8.2 

7.7 

7.2 

6.8  1  6.4 

6.2 

5.8 

80 

13.2 

12.7 

12.0 

11.3 

10.5 

9.9 

9.2 

8.7 

8.1 

7.6'  7.1 

6.6 

6.2 

100 

14.7  14.1 

13.4 

12.8 

12.0 

11.3 

10.6 

9.9 

9.1 

8.5 

7.9 

7.3 

6.8 

120 

16.2  15.4 

14.9 

14.2 

13.4 

12.7 

12.0 

11.3 

10.4 

9.8 

8.9 

8.2 

7.6 

140 

17.7  17.2 

16  4 

156 

14.9 

14  2 

13.4 

12.7 

11.9 

11.1 

10.2 

9.6 

8.8 

160 

19.1  18.0 

17.9 

17.3 

16.6 

15.7 

15.0 

14.2 

13.3 

12.6 

11.7 

10.9 

10.0 

180 

20  3 

19.9 

19.4 

18.8 

18.0 

17.3 

16.7 

15.8 

15.0 

14.1 

132 

12.4 

11.5 

200 

21.5 

21.2 

20.8 

20.2 

19.3 

18.9 

18.1 

17.5 

16.0 

15.7 

14.9 

14.0 

13.1 

220 

22.5 

223 

21.9 

21.5 

21.0 

20.3 

19.7 

19.0 

18.2 

17.5 

16.6 

155 

14.7 

240 

23.2 

230 

22.9 

22  5 

22.0 

21.6 

21.1 

20.5 

19.8 

19.1 

18.2 

17.3 

16.4 

260 

239 

238 

23.7 

23.5 

23.1 

22.7 

22  3 

21.8 

21.2 

20.6 

19.8 

19.1 

18.1 

230 

24.1 

24.3 

24.2 

24.2 

24.0 

23.7 

23.5 

23.1 

22.4 

21.8 

21.2 

20.5 

19.8 

300 

24.3 

24.5 

24.6 

24.6 

24.5 

24.4 

24.2 

23  9 

23.6 

23.1 

22.5 

21.9 

21.2 

320 

24.2 

24.5 

24.7 

24.9 

24.8 

24.8 

24.8 

24.7 

24.4 

24.1 

23.7 

23.1 

22.5 

340 

23.7 

24.2 

24.5 

24.7 

25.0 

25.2 

25.1 

25.0 

25.0 

24.9 

24.6 

24.1 

23.7 

360 

23.2 

23.7 

24.2 

24.5 

24.7 

25.0 

25.1 

25.3 

25.4 

25.3 

25.1 

24.9 

24.5 

380 

22.5 

23.1 

23.6 

24.1 

24.4 

24.7 

25.1 

25.2 

25.4 

2.).5 

23.4 

25  3 

25.2 

400 

21.5 

22.3 

22.8 

23.4 

23.9 

24.3 

24.7 

25.1 

25.2 

25.4 

25.6 

25.6 

25.5 

420 

20.6 

21.3 

22.0 

22.6 

23.1 

23.6 

24.1 

24.5 

25.0 

2^.2 

25.4 

25  6 

25.7 

440 

19.6 

20.3 

21.0 

21.8 

22  3 

22.9 

23.4 

23.9 

24.3  24.8 

25.0 

25.2 

25.6 

460 

187 

194 

20.1 

20.7 

21.3 

21.9 

22.6 

23.3 

23  6  24.1 

21.6 

24.8 

25.1 

480 

17.9 

185 

19.1 

19.7 

20.3 

21.0 

21.6 

22  2 

22.8  23.3 

23.8 

24.3 

24.6 

500 

16.9 

17.6 

18.2 

18.8 

19.3 

19.9 

20.7 

21.4 

21.9 

22.5 

22.9 

23.4 

23.9 

520 

16.2 

16.8 

17.3 

17.9 

18.4 

19.0 

19.7 

20.4 

21.0 

21.6 

21.1 

22.6 

23.0 

540 

15.4 

16.1 

16.6 

17.2 

17.5 

18.1 

18.7 

193 

19.9 

29.5 

21.2 

22.7 

22  2 

560 

14.9 

154 

10.0 

16.5 

16.9 

17.3 

17.9 

18.4 

18.9 

19.6 

20.1 

20.7 

21.3 

580 

145 

15.0 

15.3 

15.9 

16.3 

16.7 

17.1 

17.6 

18.1 

18.7 

19.3 

19.8 

20.3 

600 

14.2 

14.G 

14.  Q 

15.3 

15.8  16.3 

16.6 

17.0 

17.4 

17.9 

18.3 

18.9 

19.4 

620 

138 

14.2 

14.6 

14.9 

15.1 

15.7 

16.2 

16.6 

16.9 

17.3 

17.6 

18.0 

18.5 

640 

13.5 

14.0 

14.2 

14.6 

14.8 

15.1 

15.6 

16.1 

16.5 

16.8 

17.1 

17.5 

17.9 

660 

13  2 

13.5 

13.9 

143 

14.6 

14.9 

15.2 

15.6 

15.9 

16.4 

16.6 

17.0 

17.3 

680 

12.8 

13.2 

13  5 

13  9 

14.2 

14.5 

14.9 

15.2 

15.6 

16.0 

16.2  16.5 

16.8 

700 

12.4 

12.9 

13.3 

13.5 

13.8 

14.2 

14.5 

14.9 

15.1 

15.6 

15.9 

16.2 

16.4 

720 

12.0 

12.4 

12  8 

132 

135 

13.8 

14.2 

14.5 

14.8 

15.1 

15.5 

158 

16.1 

740 

11.5 

11.9 

12.2 

126 

12.9 

13.3 

13.8 

14.2 

14.5 

14.8 

15.1 

15.4 

15.7 

700 

10  9 

11.4 

11.8 

12.2 

12.4 

12.8 

13.2 

137 

14.1 

145 

14.7 

15.0 

15.4 

780 

10.2 

10.6 

11.2 

11.6 

11.9 

12.4 

12.8 

13.2 

13.5 

139 

14.3 

14.6 

14.9 

800 

9.5 

10.0 

10.3 

10.9 

11.3 

11.6 

12.1 

12.6 

129 

13.4 

13.8 

14.2 

14.5 

820 

8.7 

93 

97 

100 

10.5 

10.9 

11.4 

11.9 

12.3 

12.8 

13.2 

13.6 

14.0 

840 

8.1 

8.4 

88 

9.3 

9.6 

10.1 

10.6  11.1 

11.6 

12.1 

12.5 

130 

13.4 

860 

7.6 

7.9 

8.1 

8.5 

8.8 

9.2 

9.7 

,  10.2 

10.7 

11.2 

11.7 

12.1 

12.6 

880 

7.2 

7.4 

7,6 

7.8 

8.1 

8.5 

8.8 

9.4 

9.8 

10.2 

10  7 

11.2 

11.8 

900 

6.8 

7.0 

7.1 

7.3 

7.4 

7.8 

8.2 

8.5 

8.9 

9.4 

9.8 

10.3 

10.8 

920 

6.7 

6.8 

6.8 

0.9 

7.0 

7.0 

7.4 

7.8 

8.1 

8.6 

8.9 

9.4 

99 

940 

6.7 

6.7 

0.7 

68 

6.7 

6.8 

6.8 

7.1 

7.4 

7.7 

8.1 

84 

89 

960 

7.1 

7.0 

6.8 

6.7 

6.5 

6.5 

6.6 

6.7 

68 

7.1 

73 

7.7 

8.0 

980 

7.7 

7.4 

7.1 

6.9 

6.6 

6.5 

6.4 

6.4 

6.3 

65 

6.8 

6.9 

7.3 

1000 

8.4 

8.0 

7.7 

7.3 

69 

6.7 

6.5 

!  6.5 

6.3 

6.2 
7-iO 

6.2 

6.4 

740 

6.5 

63) 

n.n 

P.S1 

66  1  670  630 

690 

700 

710 

7.J> 

750 

TABLE  XXXII. 


45 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II.  ,  750  J  760  770 

1 780  790 

]800 

810  1  820 

830 

840 

,  850 

860 

870 

0 

6.5 

6.8 

7.2 

7.5 

8.0 

8.4 

8.8 

9.5 

10.1  10.5  11.0 

11,6  12.4 

20 

6.1 

6.2 

0.5 

6.7 

7.0 

7.4 

7.9 

8.4 

9.0  9.5  10.0 

10.6  11.1 

40 

5.8 

5.9 

5.9 

6.2 

6.4 

6.6 

6.9 

7.4 

7.8,  8.2 

8.8 

9.5  10.0 

60 

5.8 

5.7 

5.7 

5.7 

5.9 

6.1 

6.2 

6.5 

6.9 

7.2 

7.7 

8.3  8,8 

80 

6.2 

5.8 

5.7 

5.6 

5.4 

5.6 

5.7 

5.9 

6.1 

6.3 

6.7 

7.3 

7,8 

100 

6.8 

6.3 

5.9 

5.6 

5.5 

5.3 

5.3 

5.4 

5.4 

5.6 

5.9 

6.3 

6.8 

120 

7.6 

7.4 

6.5 

6.0 

5.7 

5.5 

5,1 

5.2 

5.1 

5.1 

5.2 

5.5 

5.8 

140 

8.8 

8.1 

7.4 

6,8 

6.2 

5.8 

5,4 

5.2 

5.0 

4.9 

4.8 

50 

5.1 

160 

10.0 

9.3 

8.5 

7.8 

7.2 

6.5 

5,9 

55 

5.1 

5.9 

4.7 

47 

4.7 

180 

11.5 

10.6 

9.7 

9.0 

8.2 

7.5 

6,9 

6.3 

5.8 

5.2 

4.8 

4.7 

4.5 

200 

13.1 

122 

11.2 

10.4 

9.5 

8.8 

7.9 

7.1 

6.5 

5.9 

6.3 

5.0 

4.7 

220 

14.7 

13.8 

12.9 

12.0 

11.1 

10.2 

9.3 

8.4 

7.5 

6.7 

6.1 

55 

5.2 

240 

16  4 

15.3 

14.5 

13.0 

12.6 

11.7 

10,7 

9.8 

8.8 

7.9 

7.0 

6.5 

5.9 

260 

18  1 

17.2 

16.3 

15.3 

14.3 

13.3 

12,2 

11.4 

10.4 

94 

8.3 

77 

6.9 

280 

19  8 

18.9 

17.9 

170 

16.1 

150 

14.0 

13.0 

11.9 

10.9 

9.9 

8.9 

8.0 

300 

21  2 

20.4 

19.0 

18.7 

17.7 

16.8 

15.8 

14.7 

13.7 

12.6 

11.5 

10.5 

94 

320 

22  5 

21.9 

21.2 

20.4 

19.4 

18.5 

17.4 

16.5 

15.5 

14,2 

13.2 

12.3 

11,2 

340 

23.7 

23.0 

22.4 

21.8 

21.1 

20  2 

19.2 

18.3 

17.1 

16.1 

15.0 

13.9 

12,9 

360 

24.5 

24,0 

23.6 

23.0 

22,4 

21.6 

20.8 

19.9 

18.9 

17.9 

16.8 

15.9 

147 

380 

25.2 

24.9 

24.5 

24,0 

23.5 

22,8 

22.1 

21.4 

20.5 

19.5 

185 

17  6 

16,5 

400 

25.5 

25.4 

25.1 

24.8 

24  5 

23.9 

23.4 

22  7 

21.9 

21.0 

20.1 

19.2 

18.2 

420 

25.7 

25.6 

25.5 

25.3 

250 

24.5 

24.2 

23.7 

23.2 

22  3 

21.5 

20.7 

19.8 

440 

25.6 

25.6 

25.7 

25.7 

25.5 

25.3 

24,9 

24.0 

24.1 

23.4 

22.7 

220 

21.2 

460 

25.1 

25.3 

25.5 

25.6 

25  8 

25.7 

25,4 

25.2 

24.8 

24.3 

X37 

23  1 

22  5 

480 

24.6 

24.9 

25.2 

25.4 

25  6 

25.6 

25  5 

25.4 

25.2 

24.9 

24.5 

24.1 

23.5 

500 

23.9 

24.2 

24.7 

25.0 

25.3 

25.4 

25.5 

25.5 

25.4 

25.2 

249 

24.7 

24.3 

520 

23,0 

23.6 

23.9 

24.3 

24.7 

24.9 

25.2 

25.4 

25.4 

25.3 

25.2 

25.1 

24.8 

540 

Oil  o 

22,(5 

23.2 

23.6 

24.0 

24.4 

24.6 

24.9 

25.1 

25.0 

25.1 

25.1 

25  0 

560 

21.3 

21,7 

22.2 

22.8 

23.2 

23.7 

24.0 

24.3 

24.6 

24.7 

24.8 

24.9 

24.9 

580 

20.3 

20.8 

21.3 

21.8 

22.3 

22  7 

23.2 

23.7 

23.9 

24.1 

24.4 

24  6 

24.7 

600 

19.4 

19.9 

20.4 

20.8 

21.4 

21.9 

22.2 

22.7 

23.1 

23.4 

23.7 

24.1 

24.3 

620 

18.5 

19.0 

19.5 

20.1 

20.5 

209 

21.4 

21.8 

22.2 

22.6 

22.9 

233 

23.6 

640 

17,9 

18.3 

18.7 

19.2 

19.7 

20,1 

20.5 

22.0 

21.3 

21.7 

22.1 

22.5 

22.8 

660 

17.3 

17.6 

18.1 

18.5 

18.9 

19.4 

19.6 

20.1 

20.5 

20.7 

21.2 

21.7 

22.0 

080 

16.8 

17.1 

17,4 

17.8 

18,2 

18.6 

18.9 

19.4 

19.7 

20.1 

20.4 

20.7 

21.2 

700 

16.4 

16.7 

16.9 

17.3 

17.7 

18.0 

18.3 

18.7 

18.9 

19.2 

19.6 

20.0 

i.0.3 

720 

16.1 

163 

16,5 

16,9 

17,2 

17.0 

17.8 

18.0 

18.3 

18.5 

18.7 

19.2 

19.5 

740 

157 

16.0 

16.2 

16.5 

107 

17,0 

17.3 

17.0 

17.8 

17.9 

18.1 

18.5 

18.8 

760 

154 

15.7 

16.0 

16  1 

16,4 

16,6 

16,7 

17.2 

17.4 

17.4 

17.8 

180 

18.2 

780 

14,9 

153 

15.6 

15.9 

16,1 

163 

16,5 

16.7 

16.9 

17.1 

17.3 

17.6 

17.7 

800 

14.5 

14,7 

15.2 

15.5 

15.8 

1.5.9 

16,2 

16.5 

16.6 

16.8 

16.9 

17.1 

17.3 

820 

14,0 

14.4 

14.7 

1.5.1 

15.4 

15.7 

15.8 

16.1 

16.3 

16.4 

16.6 

16.9 

17.0 

840 

13.4 

137 

14  1 

14.5 

15.1 

15.4 

15.4 

15.8 

15.9 

16.1 

16  2 

16.6 

10.7 

860 

12.6 

13.1 

13.5 

139 

14.3 

14.8 

15.2 

15.5 

15.6 

158 

16.0 

16.3 

16.4 

880 

11.8 

12.3 

12.8 

133 

13.7 

14.1 

14.5 

15,0 

15.3 

154 

15  6 

15.9 

16.1 

900 

10.8 

11.3 

11.9 

12.4 

13.0 

13  4 

13.7 

14,2 

14.7 

15.0 

15,2 

15.5 

15.7 

920 

99 

10.3 

10,8 

11.4 

12.0 

12.5 

12.9 

13,4 

14.0 

14,3 

14.7 

15  0 

15.3 

940 

89 

9.4 

99 

10.4 

11.0 

11.6 

121 

125 

13.0 

13,6 

13.9 

14.4 

14.7 

960 

8.0 

8.3 

88 

94 

10.0 

10.6 

11,1 

11.7 

12.2 

12,5 

13.1 

13.7 

14.1 

980 

7,3 

7  6 

7,9 

8.4 

89 

9.5 

9,9 

10  5 

11,1 

11.6 

12.1 

12.8 

133' 

1000 

6,5 

6.8 

7.2 

7.5 
780 

8.0 

8.4 

8,8 

9.5 

P20  ' 

10.0 

10  5 

11.0 

11.6 

8(0  1 

12,4 

750 

760 

770 

790  800 

810 

840  1 

P50  1 

870 

46 


TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V, 

V. 


11.      870 

880 

890 

900 

910 

920 

930  i  940 

950 

960 

970 

980 

990 

1000 

0    12.4 

12.9 

13.2 

136 

13  9 

14.2 

144 

14.8 

15  0 

15  I 

15.1 

152 

15  2 

15.3 

20    11.1 

11.7 

12.2 

12.7 

132 

13  6 

138 

14.1 

144 

14.7 

14.8 

15.0 

14.9 

14.9 

40    10  0 

10  5 

11.1 

11.7 

12.3 

126 

13  0 

13.4 

13.7 

14  1 

143 

14.6 

14.7 

147 

60      8.8 

94 

9.9 

10.6 

11.2 

11.8 

12  1 

12  6 

12.9 

133 

13  6 

139 

14.2 

14.4 

80      7.8 

8.3 

8.7 

93 

10.0 

10.5 

11.1 

11.6 

12.1 

12.5 

12.8 

13.2 

13.5 

13.8 

100      6.8 

7.2 

7.6 

8.1 

8.6 

9.4 

9.9 

105 

10.9 

11.4 

12.0 

12.4 

12.8 

132 

120      5.8 

6.1 

66 

7.1 

7.6 

8.1 

8.7 

9.4 

9.9 

104 

10.8 

11.4 

11.8 

12.3 

140      5.1 

53 

5.6 

6.0 

6.5 

7.0 

7  5 

82 

8.7 

93 

9  7 

103 

10.8 

11.3 

160     4.7 

4.8 

4.8 

5  2 

56 

5.9 

63 

6.8 

7.4 

8.0 

80 

92 

9.7 

10.2 

180 

4.5 

4.5 

4.4 

4.5 

48 

5.1 

5.4 

58 

6.2 

6.9 

7.4 

8.0 

8.4 

9.1 

200 

4,7 

4.5 

4.2 

4.2 

4.2 

4.4 

4.6 

5.0 

5.3 

5.7 

6.3 

6.9 

7.4 

7.8 

220 

5.2 

47 

43 

4.2 

41 

4.1 

40 

43 

4  5 

4.8 

5.1 

5.7 

6.2 

6.8 

240 

5  9 

5.3 

4.7 

43 

4.1 

4.0 

38 

39 

40 

4.2 

43 

4.7 

52 

5.7 

2fi!) 

6.9 

61 

54 

4.9 

44 

4.1 

3.8 

3.7 

3  6 

3.7 

38 

4.1 

43 

4.9 

280 

80 

72 

63 

5  7 

52 

4.6 

4  1 

38 

3  5 

3.5 

35 

3.6 

3.7 

3.9 

300 

9.4 

8.5 

75 

6.8 

6.1 

5.4 

4.7 

4.3 

3.9 

3.6 

3.3 

3.3 

3.3 

3.4 

320 

112 

101 

9.1 

8.1 

73 

6.5 

5.7 

50 

4.4 

40 

36 

34 

32 

3.2 

340 

12.9 

11.8 

107 

9.6 

8.7 

7.7 

6.8 

60 

5.2 

4.6 

4.1 

3.7 

3.4 

3.2 

360 

14.7 

13.4 

12  3 

11.1 

10.1 

9.2 

83 

7.4 

6.4 

5.7 

49 

4.3 

38 

3.5 

380 

16  5 

15.4 

14.2 

13  0 

11.8 

108 

97 

8.7 

78 

6.9 

61 

5.4 

46 

4.1 

400 

18.2 

17.2 

16.0 

14.9 

13.8 

12.4 

11.4 

10.4 

93 

8.3 

7.3 

6.4 

56 

5.0 

420 

19  8 

18.8 

17.7 

16  7 

15.5 

144 

13.1 

11.9 

10  9 

9.8 

8.8 

8.0 

6.9 

0  1 

440    21.2 

2f)3 

103 

18.3 

17.3 

162 

14  9 

13  8 

12.7 

11.5 

10.5 

95 

84 

7.5 

460 

22  5 

21  6 

20  n 

19.7 

189 

179 

16.7 

15.6 

143 

13  3 

12.2 

10.9 

100 

90 

480 

23  5 

22  7 

22  0 

21.1 

20  2 

19  3 

18  2 

17.3 

16  2 

15.0 

13.8 

12  8 

11.6 

10  5 

500 

24.3 

23.8 

230 

223 

21.6 

20.7 

19.7 

18.8 

17.8 

16.7 

15.4 

145 

13.4 

12.3 

520 

248 

24  3 

23  7 

23  2 

22.7 

21.9 

21.1 

20.2 

19  2 

18.3 

17.2 

16.1 

15  0 

14.0 

540 

25.0 

24  8 

24  3 

23  9 

23.4 

22  8 

22  1 

21.3 

20.6 

19.7 

18.7 

17  6 

16.6 

15.6 

560 

24  9 

24.8 

24  7 

24-4 

24.0 

23  6 

22  9 

22  4 

21.6 

20.8 

20.0 

19.1 

18.2 

17.1 

580 

24  7 

24.7 

246 

24.5 

24  3 

23  9 

23.5 

23  1 

22.5 

21  9 

21.1 

20.3 

19.5 

18.6 

600 

24  3 

24  3 

243 

24.3 

24.3 

24.1 

238 

23.5 

23.0 

22  5 

22.0 

21.4 

20.6 

198 

620 

236 

23  7 

23  9 

24.0 

24.1 

24.1 

23.9 

237 

23  4 

23.1 

22.6 

22.1 

21.4 

20.8 

640 

22  8 

23.1 

23  2 

23.4 

236 

23  7 

23.8 

23  7 

23.5 

23.2 

22.9 

22  6 

22.1 

21.6 

660 

22.0 

223 

22  5 

22.8 

23  0 

23  2 

232 

23  3 

232 

23  1 

23  0 

22.8 

22.5 

22.1 

6S0 

21.2 

21  5 

21.7 

22  0 

223 

22  .5 

22.6 

22.8 

22.9 

22.9 

22.8 

22  7 

22.7 

22.3 

700 

20.3 

20.7 

20.9 

21.2 

21.5 

21.7 

21.9 

222 

22.3 

22.5 

22.5 

225 

22.4 

22.2 

720 

19.5 

198 

20.1 

20.4 

20.8 

21  1 

21.2 

21  4 

21.6 

21.8 

21  9 

22.0 

22  0 

22.0 

740 

188 

190 

19  2 

19.6 

19.9 

20  2 

20  5 

20  7 

20  9 

21.1 

21.2 

21.5 

21.5 

21.6 

760 

18  2 

185 

18  4 

18  8 

19  1 

1.1  4 

19.0 

19.9 

20  1 

20  3 

20  5 

208 

21.0 

21.2 

780 

17.7 

17.8 

ISO 

18.1  :  184 

18  7 

18  8 

19  1 

19  3 

19  5 

19  7 

20.0 

20  2 

20.4' 

800 

17.3 

17.4 

17.4 

17.7 

179 

18.0 

18.1 

18.4 

18.6 

18.9 

IS. 9 

19.1 

19.4 

19.0 

820 

170 

17.2 

172 

17.2 

174 

174 

17.6 

178 

17.8 

18.1 

18.3 

18.5 

18.6 

18.8 

840 

167 

IfiS 

168 

169 

172 

17  2 

17.1 

17  1 

173 

17  4 

17  5 

17.8 

17.9 

18.1 

860 

164 

16  5 

165 

166 

16  r, 

16  7 

168 

16.9 

16  9 

170 

17.0 

17.1 

172 

17  4 

880 

16  1 

163 

16  3 

16.5 

16  5 

16.5 

16.6 

16  6 

16.6 

16  0 

16.6 

10.7 

10.7 

169 

900 

15  7 

15.9 

16.1 

162 

163 

16.4 

16.3 

16.3 

16.2 

16.2 

16.2 

16.3 

16.3 

16.3 

920 

15  3 

15  5 

15  6 

15  9 

16.0 

16.1 

16.1 

16.1 

160 

16  1 

16.1 

16.1 

16.0 

160 

940 

14.7 

15  9 

15  2 

15  4 

15  7 

15  8 

15  8 

160 

15  9 

15  9 

15.9 

15.8 

15  7 

15.8 

960 

14  1 

14  3 

14  5 

148 

15  2 

15.5 

15.5 

15  7 

15  7 

15  7 

15.6 

15  6 

15  5 

155 

980 

133 

12  7 

139 

142 

14  5 

14  8 

15.1 

15.3 

15  4 

15.5 

15.4 

15.4 

15.4 

15.3 

1090 

12.4 
870 

12.9 
88:) 

132 

13.6 

13.9 
910 

14.2 
920 

14.4 

14  8 
940 

15.0 

15.1 

15.1 

15.2 
980 

15.2 
990 

15.3 
1000 

890 

9T) 

930 

950 

9G0 

970 

TABLE  XXXIII. 

Perturbations  p?uduced  by  Saturn. 

Arguments  II  and  VII. 
VII. 


47 


II 

0 

100 

200 

300 

400 

500 

600 

700 

0.5 
0.7 
0.8 
1.0 
1.1 
1.3 

1.1 
0.8 
0.3 
0.2 
0.5 

800 

900 

1000 

0 

100 
200 
300 
400 
500 

600 
700 
800 
900 
1000 

1.2 
0.9 
0.7 
0.9 
1.0 
l.I 

1.2 
1.4 
1.6 
1.5 
1.2 

1.5 
1.2 
0.9 
0.8 
0.9 
1.0 

1.1 
1.1 
1.3 
1.4 
1.5 

1.4 
1.3 
1.0 
0.7 
0.6 
0.8 

0.9 
1.0 
1.0 
1.1 
1.4 

1.0 
1.1 
1.1 
0.8 
0.4 
0.4 

0.6 
0.8 
0.8 
0.9 
1.0 

0.7 
0.9 
1.0 
0.9 
0.6 
0.2 

0.2 
0.4 
0.6 
0.7 
0.7 

06 
0.8 
0.9 
1.0 
0.9 
0.5 

0.2 
0.1 
0.4 
0.6 
0.6 

0.5 
0.7 
0.8 
1.0 
1.0 
1.0 

0.5 
0.3 
0.1 
0.3 
0.5 

0.4 
0.6 
0.9 
1.0 
1.1 
1.3 

1.5 
1.4 
1.0 
0.6 
0.4 

0.8 
0.7 
0.8 
1.0 
1.1 
1.2 

1.5 
1.7 
1.6 
1.2 
0.8 

1.2 
0.9 
0.7 
0.9 
1.0 
1.1 

1.2 
1.4 
1.6 
1.5 
1.2 

Constant,  l."0 


TABLE  XXXIV. 

Variable  Part  of  Sun's  Aberration. 
Argument,  Sun's  Mean  Anomaly. 


Os 

Is 

Us 

Ills 

TVs 

Vs 

o 
0 

0.0 

0.0 

0.1 

0.3 

0.5 

0.6 

0 

30 

3 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

27 

6 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

24 

9 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

21 

12 

0.0 

0.1 

0.2 

0.4 

0.5 

0.6 

18 

15 

0.0 

0.1 

0.2 

0.4 

0.5 

0.6 

15 

18 

0.0 

0.1 

0.2 

0.4 

0.5 

0.6 

12 

21 

0.0 

0.1 

0.3 

0.4 

0.6 

0.6 

9 

24 

0.0 

0.1 

0.3 

0.4 

0.6 

0.6 

6 

27 

0.0 

0.1 

0.3 

0.4 

0.6 

0.6 

3 

30 

0.0 

0.1 

0.3 

0.5 

0.6 

0.6 

0 

Xls 

x^- 

IX^ 

VIIIs 

VIIs 

VJs 

Constant,  0."  3 


48 


TABLE  XXXV. 
Moon's  Epochs. 


Years. 

1 

2 

3  1  4    5  j  6    7 

8 

9 

10 

11 
226 

12 

458 

13 
468 

1830 

00174 

4541 

4461  4638  9885  0635*5979  J9921  7623 

219 

1831 

00103 

1749 

4127  9381  2357|6432  7040,2378  6487 

825 

587 

177 

940 

1832  B 

00032  8957 

37934125  4829,2229  8100|4835  5351  ^432 

948 

897 

413 

1833 

00235  6816 

4499  9156 

7636  8399  9219  7683  4239  108 

340 

687 

920 

1834 

00164  4024 

4164 

3900 

0107  4196  0279  0140  3103  715 

701 

406 

393 

1835 

00093  1232 

3830 

8644 

2579  9993  1340  2598  1967  321 

061 

125 

866 

1836  B 

00022  8441 

3496 

3388  5051^5791  2400  5055  0831^928 

422 

845 

339 

1837 

00224  6299 

4202 

841917858  1960  3518J7903  97191605 

814 

635 

846 

1838 

00153 j 3508 

3868 

3163:0329  7757  4579;03G0  8583:211 

175 

354 

319 

1839 

00082  0716 

3534 

7907:2801 

3555  5639 

2818  7447  818 

536 

074 

792 

1840  B 

00011  7925:3199 

2651  5273 

1 

9352  6700 

1 

5275  6310 

424  896 

1 

793 

265 

1841 

00213  5783  3906 

7682  8080 

5522  7818 

8123  5199 

101  288 

583 

772 

1842 

00142  2991  3571 

2425  0551 

1319  8879 

0580  4062 

707  649 

302 

245 

1843 

00071  0200  3237 

7169  3023 

7116  9939 

3038:2926'314  010 

022 

718 

1844  B 

00000  740S  2903 

1913,5495 

2914  1000 

5495  1790  920,371 

741 

191 

1845 

00203 

5266  3609 

6944  8302 

9083  2118  8343  0678  597 

763 

531 

698 

1846 

00132 

2475 '3275 

1688  0773 

4880  3179  0800  95421203 

123 

250 

171 

1847 

00061 

9683  2941 

6432*3245 

0678  4239 

3257  8406  810 

484 

970 

644 

1848  B 

99990 

6892  2606 

1176 

5717 

6475;5300 

5715  7270  416 

845 

689 

117 

1849 

00192 

4750;3312 

6207 

8524 

2644  6418 

8563  6158  093 

237 

479 

624 

1850 

00121 

1958  2978 

0951 

0995 

8442  7479 

1020  5022]700 

597 

199 

097 

1851 

00050 

91672644 

5695 

3467 

4239 

8539  3477 '3885 

306 

958 

918 

570 

1852  B 

99979 

63752310 

04395939 

0036 

9600  5935  2749 

913 

319 

637 

043 

1853 

00181 

4233  3016 

5469  8746 

6206 

0718  8782  1637 

589 

711 

427 

550 

1854 

00110 

1442  2681 

0213  1217 

2003 

1778' 1240  0501!  196 

072 

147 

023 

1855 

00039 

8650  2347 

1 

4957  3689 

7801  2839  3697  9365J802 

432 

866 

496 

1856  B 

99968 

5859  2013 

9701  6160 

3598  3899  6155  82291409 

793 

586 

969 

1857 

00171 

3717  2719 

4732  8968 

9767  5018  9002  7117  086 

185 

375 

476 

1858 

00100 

0925 

2385 

9476 

1439 

5565,6078' 1460  5981  692 

546 

095 

949 

1859 

00029 

8134 

2051 

4220 

3911 

1362  7139:3917  4845  299 

907 

814 

422 

1860  B 

99958 

5342 

1716 

8964 

6383 

7159  8199  6374  3709 

905 

267 

534 

895 

1861 

00160 

3200 

2423 

3995 

9190 

3329  9317i9222  2597 

581 

659 

323 

402 

1863 

00089 

0409 

2088 

8739 

1661 

9126  0378  16791461 

188 

020 

043  875 

1863 

00018 

7617 

1754 

34834133 

4923' 1438  4137  0324 

795 

381 

762  348 

1864  B 

99947 

4826 

1420 

82276605 

0721 

2499  6594  9188 

401 

742 

482 

821 

1865 

00149 

2684 

2126 

3257 

9412 

6890 

3617,9442  8076 

078 

134 

272 

328 

1866 

00078 

9893 

1792 

8001 

1883 

2687 

4678' 1899  6940 

685 

494 

991 

801 

1867 

00007 

7101 

1457 

2745 

4355 

8485 

5738 '4357,5804 

291 

855 

711 

274 

1868  B 

99936 

4309 

1123 

7489 

6827 

4282 

6799  6814  4668 

898 

216 

431 

747 

1869 

00138 

2168 

1829 

2520 

9634 

0452 

7917  9662  3556 

574 

608 

220 

254 

1870 

00067 

9376 

1495 

7264l2105|6249 

8978  2119  2420 

181 

968 

940 

727 

TABLE  XXXV. 
Moon's  Epochs. 


49 


Years. 


1830 

1831 

1832  B 

1833 

1834 

1835 

1836  B 
1837 
1838 
1839 
1840  B 

J841 
1842 
1843 
1844  B 
1845 

1846 
1847 
1848  B 
1849 
1850 

1851 
1852  B 
1853 
1854 
1855 

1856  B 
1857 
1858 
1859 
1860  B 

1861 
1862 
1863 
1864  B 
1865 

1866 
1867 
1868  B 
1869 
1870 


14 


921 
115 
309 
602 
796 
989 


15 


16 


17 


392  230  588 
532  589  940 


673,949 
844|345 
984,704 
124:063 


183  265  423 


476 
670 
864 
058 

351 
544 
738 
932 
225 

419 
613 
806 
099 
293 

487 
681 
974 
168 
361 

555 
848 
042 
236 
430 

723 
916 
110 
304 
597 

791 
985 
178 
471 
665 


493  960 
633  319 


436819 
576  178 
716  537 
857  897.197 


028 
168 
308 
449 
620 

760 
901 
041 
212 
352 


293 
688 
040 
393 

745 
140 
492 

845 


293  592 
652  944 
012  297 
371  649 
767  044 


18 


46 
937 

412 
913 
388 
863 

338 
840 
315 
790 


19 


20  21  22,23  24  25  26  27  28  29  30  31 


523  536 
296  703 

070  870 
845,037 
619  203 
392  370 


166 
942 
715 
489 
265  262 


766 
241 
716 
191 
692 


038 
811 
585 
358 
134 


537! 
704, 
870, 
037J 

204 

371! 
537 
704 

871 ; 

038 ! 


5260'44 
307041 

07,81  38 
85  92  45 
62 1 03  42 
39  13  38 

17j24  35 
94135  42 
72  46  38 
56  35 
6732 


126  396  167  907  204191 
486  749  643  68l|371j,68 
845  101  118|454;538|45 
241  496,619  2301705 '23 
600  848  094  003:87r00 


904 
944 
085 

225 
390 
537 
677 
817 

988 
129 
269 
409 
580 

721 
861 
001 
172 
313 


20l!569!777 
5o3|044|550 
948  545  326 
300  020  099 
653  495873 


038,78 
205  55 


005 
400 
752 
105 
457 


970  646 

471j422 
947  195 
422  969 

8971742 


715 
074 
434 

793 

189 
548 
908 
267 

663 

022 

382 

741 

137 

I   I 
496  657  799,387 

856  009:274:161 

215 

611 

970 


852  3981518 
204J873  291 
557348|065. 
909  823  838 
304  324614 


362 
756 
109 


749  934 
251  710 
7261483 


j37 

539 

705 

872 
039 
206 
372 
539 

706 
873 
039 
206 
373 

540 
707 
873 
040 
207 


I 
78  39 
89  35 
99  32 
10  29 
21  36 


07  61 
63 


48 

19 
90 
60 
31 
02 

73 
43 
14 

85 
56 

32  26 

29  97 
26,68 

33  39 
2909 

2680 
23|51 

30  22 

J26;9: 
|23  63 

20  34 
27  05 
24  76 
20  46 
17il7 

24  88 
20  60 
17  29 
14  00 

21,71 

17  42 
1412 
11  83 

18  54 
15'26 


98 
48 
97 
92  53 
40  03 
87  51 

34  01 

85  58 
32'07 
80'56 
2706 


99  99  89 

24  24'51 
48!49ll4 
7777,77:27 
0L0l{39  18 
26  26,02  id 


81  81 
06  06 
3031 
55  55 

84  83 


17  16:08  08 


22  03 
24I5O 


65  33  33 

15  57  58 
71  86  86 

20  io!io 

I    ! 

70  35  35  02  75 
19  59  60  65  66 

76  88  88  28  58 
25  12  12  90'50 
74  37,37  53141 


64  01 
2793 
89,84 
52  76 
14,67 


61  62 

90  90 
15  15 
39'40 
64|64 

9292 

17,17 
41*42 
66  66 
95^94 


38  19  19  41  49 


87  44  44 
37  68  69 
93  97197 
43,2ll21 


15  33 
78  24 
40' 15 
03|07 
65  99 

28  91 
91  82 
53:74 
16'65 

7857 


03  40 
66;32 
28  23 
91  15 


G 


50 


TABLE  XXXV. 
Maori's  Epochs. 


Years. 

Evection. 

Anomaly. 

Variation. 

Longitude. 

1830 

s 
5 

O    '    " 

17  4  12 

s 
11 

o 
24 

31 

4.5 

s 
2 

o 

13 

2 

39 

s 

11 

o 

22 

55 

37.7 

1831 

11 

7  35  41 

2 

23 

14 

24.6 

6 

22 

40 

4 

4 

2 

18 

42.8 

1832  B 

4 

28  7  11 

5 

21 

57 

44.4 

11 

2 

17 

28 

8 

11 

41 

48.0 

1833 

10 

29  57  40 

9 

3 

44 

58.5 

3 

24 

6 

21 

1 

4 

15 

28.4 

1834 

4 

20  29  11 

0 

2 

28 

18.5 

8 

3 

43 

45 

5 

13 

38 

33.6 

1835 

10 

11   0  40 

3 

1 

11 

38.6 

0 

13 

21 

10 

9 

23 

1 

38.8 

1836  B 

4 

1  32  9 

5 

29 

54 

58.7 

4 

22 

58 

34 

2 

2 

24 

44.0 

1837 

10 

3  22  39 

9 

11 

42 

12.8 

9 

14 

47 

27 

6 

24 

58 

245 

1838 

3 

23  54  9 

0 

10 

25 

32.9 

1 

24 

24 

51 

11 

4 

21 

29.8 

1839 

9 

14  25  38 

3 

9 

8 

53.1 

6 

4 

2 

16 

3 

13 

44 

35.0 

1840  B 

3 

4  57  8 

6 

7 

52 

13.2 

10 

13 

39 

42 

7 

23 

7 

40.4 

1841 

9 

6  47  37 

9 

19 

39 

27.5 

3 

5 

28 

33 

0 

15 

41 

20.9 

1842 

2 

27  19  7 

0 

18 

22 

47.6 

7 

15 

5 

58 

4 

25 

4 

26.2 

1843 

8 

17  50  37 

3 

17 

6 

7.9 

11 

24 

43 

23 

9 

4 

27 

31.6 

1844  B 

2 

8  22  7 

6 

15 

49 

28.1 

4 

4 

20 

48 

1 

13 

50 

37.0 

1845 

8 

10  12  36 

9 

27 

36 

42.5 

8 

26 

9 

40 

6 

6 

24 

17.5 

1846 

2 

0  44  6 

0 

26 

20 

2.8 

1 

5 

47 

5 

10 

15 

47 

23.0 

1847 

7 

21  15  35 

3 

25 

3 

23.2 

5 

15 

24 

30 

2 

25 

10 

28.3 

1848  B 

1 

11  47  5 

6 

23 

46 

43.5 

9 

25 

1 

55 

7 

4 

33 

33.7 

1849 

7 

13  37  35 

10 

5 

33 

57.9 

2 

16 

50 

47 

11 

27 

7 

145 

1850 

1 

4  9  4 

1 

4 

17 

18.3 

6 

26 

28 

12 

4 

6 

30 

19.9 

1851 

6 

24  40  35 

4 

3 

0 

38.6 

11 

6 

5 

37 

8 

15 

53 

25.4 

1852  B 

0 

15  12  5 

7 

1 

43 

59.2 

3 

15 

43 

3 

0 

25 

16 

31.0 

1853 

6 

17  2  34 

10 

13 

31 

13.7 

8 

7 

31 

54 

5 

17 

50 

11.6 

1854 

0 

7  34  4 

1 

12 

14 

34.1 

0 

17 

9 

20 

9 

27 

13 

17.2 

1855 

5 

28  5  33 

4 

10 

57 

54.7 

4 

26 

46 

44 

2 

6 

36 

22.7 

1856  B 

11 

18  37  3 

7 

9 

41 

15.2 

9 

6 

24 

10 

6 

15 

59 

28.2 

1857 

5 

20  27  33 

10 

21 

28 

29.8 

1 

28 

13 

2 

11 

8 

33 

9.1 

1858 

11 

10  59  2 

1 

20 

11 

50.3 

6 

7 

50 

27 

3 

17 

56 

146 

1859 

5 

1  30  33 

4 

18 

55 

10.9 

10 

17 

27 

53 

7 

27 

19 

20.1 

1860  B 

10 

22  2  3 

7 

17 

38 

31.4 

2 

27 

5 

18 

0 

6 

42 

25.8 

1861 

4 

23  52  32 

10 

29 

25 

46.1 

7 

18 

54 

10 

4 

29 

16 

6.6 

1862 

10 

14  24  2 

1 

28 

9 

6.6 

11 

28 

31 

35 

9 

8 

39 

12.2 

1863 

4 

4  55  32 

4 

26 

52 

27.3 

4 

8 

9 

1 

1 

18 

2 

17.9 

1864  B 

9 

25  27  2 

7 

25 

35 

48.0 

8 

17 

46 

25 

5 

27 

25 

23.5 

1865 

3 

27  17  31 

11 

7 

23 

2.7 

1 

9 

35 

18 

10 

19 

59 

4.3 

1866 

9 

17  49  2 

2 

6 

6 

23.3 

5 

19 

12 

43 

2 

29 

22 

10.1 

1867 

3 

8  20  31 

5 

4 

49 

44.0 

9 

28 

50 

9 

7 

8 

45 

15.7 

1868  B 

8 

28  52  2 

8 

3 

33 

4.7 

2 

8 

27 

34 

11 

18 

8 

21.4 

1869 

3 

0  42  33 

11 

15 

20 

19.6 

7 

0 

16 

26 

4 

10 

42 

23 

1870 

< 

8 

21  14  2 

2 

14 

3 

40.3 

11 

9 

53 

51 

8 

20 

5 

8.0 

TABLE  XXXV. 


61 


Mooii's  Epochs. 


Years. 

Supp.  of  Node. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 
025 

XII 
433 

1830 

6  7  7  11.0 

10  24  46 

498 

502 

900 

904 

427 

062 

1831 

6  26  26  53.3 

2  15  18 

912 

914 

208 

210 

506 

001 

211 

710 

1832  B 

7  15  46  35.5 

6  5  50 

326 

327 

516 

516 

586 

940 

397 

986 

1833 

8  5  9  28.4 

10  731 

774 

779 

852 

856 

702 

885 

624 

297 

1834 

8  24  29  10.7 

1  28  3 

187 

191 

159 

163 

782 

825 

810 

573 

1835 

9  13  48  53.0 

5  18  35 

601 

603 

467 

469 

861 

764 

996 

850 

1836  B 

10  3  8  35.2 

9  9  8 

015 

016 

775 

775 

941 

703 

182 

127 

1837 

10  22  31  28.1 

1  10  49 

463 

468 

111 

116 

057 

648 

409 

437 

1838 

11  11  51  10.4 

5  1  21 

876 

880 

419 

423 

137 

588 

595 

714 

1839 

0  1  10  52.6 

8  21  53 

290 

292 

726 

729 

217 

527 

781 

991 

1840  B 

0  20  30  34.9 

0  12  25 

704 

705 

034 

035 

296 

466 

967 

268 

1841 

1  9  53  27.7 

4  14  6 

152 

157 

370 

375 

412 

411 

194 

578 

1842 

1  29  13  10.0 

8  4  38 

566 

569 

678 

682 

492 

350 

380 

855 

1843 

2  18  32  52.2 

11  25  10 

980 

980 

986 

988 

572 

290 

566 

131 

1844  B 

3  7  52  34.5 

3  15  42 

393 

394 

293 

294 

651 

229 

752 

408 

1845 

3  27  15  27.4 

7  17  23 

840 

846 

629 

634 

767 

174 

979 

718 

1846 

4  16  35  9.6 

11  7  55 

254 

258 

937 

941 

847 

113 

165 

995 

1847 

5  5  54  51.8 

2  28  27 

668 

670 

245 

247 

927 

053 

351 

272 

1848  B 

5  25  14  34.1 

6  18  59 

082 

083 

553 

553 

006 

992 

537 

549 

1849 

6  14  37  27.0 

10  20  40 

531 

535 

889 

893 

122 

937 

764 

859 

1850 

7  3  57  9.2 

2  11  12 

944 

947 

196 

200 

202 

876 

950 

136 

1851 

7  23  16  51.5 

6  1  44 

358 

359 

504 

506 

282 

816 

136 

413 

1852  B 

8  12  36  33.6 

9  22  17 

772 

772 

812 

812 

362 

755 

322 

689 

1853 

9  1  59  26.5 

1  23  58 

220 

223 

148 

152 

477 

700 

549 

000 

1854 

9  21  19  8.8 

5  14  30 

634 

636 

456 

459 

557 

639 

735 

276 

1865 

10  10  38  51.1 

9  5  2 

047 

048 

763 

765 

637 

579 

921 

553 

1856  B 

10  29  58  33.3 

0  25  34 

461 

461 

71 

071 

717 

518 

107 

830 

1857 

11  19  21  26.2 

4  27  15 

909 

912 

407 

411 

832 

463 

334 

140 

1858 

0  8  41  8.4 

8  17  47 

323 

325 

715 

718 

912 

402 

520 

417 

1859 

0  28  0  50.7 

0  8  19 

736 

737 

023 

024 

992 

342 

706 

694 

1860  B 

1  17  20  32.9 

3  28  51 

150 

150 

330  330 

072 

281 

892 

971 

1861 

2  6  43  25  8 

8  0  32 

598 

601 

666 

670 

187 

226 

119 

281 

1862 

2  26  3  8.0 

11  21  4 

012 

014 

974 

977 

267 

165 

305 

558 

1863 

3  15  22  50.1 

3  11  36 

426 

426 

282 

283 

347 

105 

491 

834 

1864  B 

4  4  42  32.3 

7  2  8 

839 

839 

590 

589 

427 

044 

677 

111 

1865 

4  24  5  25.2 

11  3  49 

287 

291 

926 

929 

542 

989 

904 

422 

1866 

5  13  25  7.3 

2  24  21 

701 

703 

233 

236 

622 

928 

090 

698 

1867 

6  2  44  49.5 

6  14  53 

115 

115 

541 

542 

702 

868 

276 

975 

1868  B 

6  22  4  31.7 

10  5  26 

529 

528 

849 

848 

782 

807 

462 

252 

1869 

7  11  27  24.6 

2  7  7 

977 

980 

185 

188 

897 

752 

689 

562 

1870 

8  0  47  6.7 

5  27  39 

390 

392 

493 

495 

977 

691 

875 

839 

52 


TABLE  XXXVI. 

Moori's     Motions  for  Mo?iths. 


Months. 


January 
February 

March 


April 
May 
June 

July 
Aug. 
Sept. 

Oct. 
Nov. 
Dec. 


Com. 
Bis. 

Com. 


{b 


Com. 


{fiis 


(  Com. 
^Bis. 
(  Com. 
i  Bis. 
(  Com. 
tBis. 

<  Com. 
iBis. 
(  Com. 
\  Bis. 
(  Com. 
i  Bis. 


00000 
08487 
16153 
16427 

24640 
24914 
.32853 
33127 
41340 
41614 

49554 
49828 
58041 
.58315 
66528 
66802 

74741 
7.5015 
83228 
8350.. 
91442 
91716 


0000  0000  0000  0000 
0146  2246  8896'0402 


8343  1371  6931 
8993  2411  7218 


9797 
0132 


8490  3616  5827  0199 
9140  4657  6114'0534 
7986  4822 '4436; 0265 
8636, 5862  4723  0600 
8133  70(.7i3332[0666 
8783  810713619  1002 


7629 
8279 
7776 
8426 
7922 
8572 

7419 
8069 
7565 
8215 
7062 
7712 


8273  194210732  7341  0444 
9313i2228il068  7713  0502 


0000  0000 
1.533: 1789 
1951 ! 3404 
2323  3462 

3484  5193 
3856  5251 
4646 1 6924 
.5018  6982 
6179:8713 
655li8771 


0518 
15.58 
2764 
3804 

3969 
5009 
6215 
7255 
7420 
3460 


0838 
1125 
9734 
0021 

8343 

8630 

7239 

7526 1 23.39 

.584812070 

613512405 


1134  8874:2233 
1470'9246J2290 
1536  0408J4021 
1871 J0780  4079 

1602' 1569  5752 

1938|l941i5810 

200413102  7541 

347517599 

426419272 

4636  9330 


10  II  I  12  13 


oooo'oooo'ooo 

2099  07.53  175 


3027 
3418 

5126 
5517 
6835 
7226 


1433 
1457 

2186 
2210 
2914 
29.38 


139 
209 

314 
384 
419 
489 
593 
663 

698 
768 
873 
943 
048 
118 

65.50  6630  152 
6941  6654' 222 
8649  7382  327 
9040  7407,397 
03.58  8111  432 
0749,8135502 


8934  3667 
9325  3691 

0643  4396 
1034  4420 
2742  5148 
31335173 
4842  5901 
5232  5925 


OOOiOOO 
9651184 
836! 157 

868|228 

80l!342 
832|412 
735,456 
766  526 


700 
731 

634 
665 
599 
630 
563 
595 

497 
528 
462 
493 
396 
427 


640 
710 

754 

824 
938 
009 
123 
193 

237 
307 
421 
492 
535 
606 


000 
059 
016 
050 

076 
110 
101 
135 
160 
194 

185 
219 
245 
279 
304 
338 

329 
363 
38S 
42.3 
414 
448 


TABLE  XXXVL 


Moon's  Motions  for  Months. 


Months. 


[January 
iFebruaxy 


Com. 


f^'"^.  \  B.s 


April, 
May 
■Jun? 

July 
Aug. 
Sept. 

lOct. 

pS'ov. 
Dec. 


<  Com. 
\.  Bis. 
(  Com. 
t  Bis. 
(  Com. 
^Bis. 

(  Com. 
\  Bis. 
(  Com. 
IBss. 
(  Com. 
t  Bis. 

5  Com. 
\  Bis. 
(  Com. 
I  h  s. 
(  Com. 
i  Bis. 


Evection. 


0  0  0  0 

11  20  48  42 

10  7  40  26 

10  18  59  26 

9  28  29  8 

10  9  48  8 

9  7  58  51 

9  19  17  50 

8  28  47  33 

9  10  6  33 

8  8  17  16 

8  19  36  15 

7  29  5  59 

8  10  24  58 

7  19  54  41 

8  1  13  40 

6  29  24  24 

7  10  43  23 

6  20  13  6 

7  1  32  5 

5  29  42  49 

6  11  1  48 


Anomaly. 


0  0  0  0.0 

1  15  0  53.1 

1  20  50  4.2 

2  3  53  58.2 


5  50  57.3 
18  54  51.2 

7  47  56.4 
20  51  .50.3 
22  48  49.4 

5  .52  43.4 


6  24  45  48.5 

7  7  49  42.5 

8  9  46  41.6 

8  22  50  3.5.5 

9  24  47  34.6 
10.  7  51  28.6 

10  26  44  33.7 

11  9  48  27.7 
0  11  45  26.8 

0  24  49  20.7 

1  13  42  25.9 
1  26  46  19.8 


Variation. 

s  o   '  " 

0  0  0  0 

0  17  54  48 

11  29  15  15 

0  11  26  42 

0  17  10  3 

0  29  21  29 

0  22  53  24 

1  5  4  50 

1  10  48  11 

1  22  59  38 

1  16  31  32 

1  28  42  59 

2  4  26  20 

2  16  37  47 

2  22  21  7 

3  4  32  34 

2  28  4  28 

3  10  15  55 

3  15  .59  16 

3  28  10  43 

3  21  42  37 

4  3  54  4 

Longitude. 

s     °     ' 

0  0  0  0.0 

1  18  28  5.8 

1  27  24  26.6 

2  10  35  1.6 

3  15  52  32.5 

3  29  3  7.5 

4  21  10  3  3 

5  4  20  38.3 

6  9  38  9.1 

6  22  48  44.1 

7  14  55  39.9 
7  28  6  15.0 
9  3  23  45.8 
9  16  34  20.8 

10  21  51  51.6 

11  5  2  26.7 

11  27  9  22.4 

0  10  19  57.5 

1  15  .37  28.3 

1  28  48  3.3 

2  20  54  59.1 

3  4  5  34.1 


TABLE  XXXVI. 
Moon's     Motions  for  Months. 


53 


Months. 


January 
February 

March 


April  i 

May  < 

June  < 

July  { 

Aug.  J 

Sept.  i 

Oct. 
Nov. 
Dec. 


Com. 
Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 


14  1 15    16 


OOOiOOO 
074  946 
8511801 
950j831 

9251747 
024778 
899  663 
999  693 


973 
073 

948 
047 
022 
121 
096 
195 

071 
170 
145 
244 
120 
219 


609 
639 

525 
555 
471 
501 
417 
447 

333 

363 
279 
309 


000 
13.5 
159 
19.6 

294 
331 
392 
429 
527 
563 

625 
661 
759 
796 
894 
931 

992 
029 
127 
163 
194)225 
225  261 


17  [  18  I  19  I  20  121122123 


OOOjOOOOOO 

304  805  066, 0141 24'26|14 
482  532  125027, 4550  98 
524  558,127  027  46,51  08 


000  ooloo'oo 

^1- 


24  25  26,27:28  29  30  31 


00  00 

28 


786  3361191 
828  362193 


041 
042 


047 
089 
351 
393 

613 
655 
917 
959 
221 
263 

483  087  578 
525  113  581 
787;892  644 
829 '918  646 
049,670  708 
091|696  710 


115  254 
141j256 
920  320 
946  322 

I 
699  384 
725  386 
503  449 
529  451 
308  515 
334  517 


68  77 

69  77 
055191  [02 
055: 92  03 
069  15,28 
069  15I29 


083 
083 
097 
097 
111 
111 


125  07 


3754 
38  55 


126 
139 
140 
153 
153 


00  00;  00 

14'l7  29 
18!l2  46 
21I19  51 


39  70;32'29  76 
4274  36  36  80 
19j94'43  38  01 
22i98,47  45  05 


65  26 

75128 


90 


21  57 
2561 

45'68 
49^72 
72  82 
7786 
00  97 
04  01 

I 
23  08 
28  11 
51  22 
55  26 
74  33 
79  37 


55  31 
62  35 

64  56 
71  60 
81  85 
88  90 
97  15 
04  19 

07  40 
14  44 
23  70 
30  74 
33  95 
40  99 


15  23 
16|23 
21  30 
21  30 
26  38 
26  38 

3145 
31^46 
36  53 
36  53 
42  61 
42  61 

I 
47  68 
47  69 
52  76 
52  76 
57  84 
57  84 


TABLE  XXXVL 

Moon's  Motions  for  Months. 


Months. 


January 
February 

March 


April 
May 
June 

July 
Aug. 
Sept. 

Oct. 
Nov. 
Dec. 


Com. 
Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 


Supp.  of  Node. 


0  0     0  0.0 

0  1  38  29.7 

0  3     7  27  5 

0  3  10  38.2 


4  45  57.3 
4  49  7.9 
6  21   16.4 

6  24  27.0 

7  59  46.1 

8  2  56.7 


0  9  35     5.2 

0  9  38   15.9 

0  11  13  350 

0  11  16  45.6 

0  12  52     4.7 

0  12  55  15.4 

0  14  27  23.8 

0  14  30  34.4 

0  16  5  53  5 

0  16  9     4.2 

0  17  41   12.6 

0  17  44  23.3 


II 


0  0  0 

11  15  43 

9  27  59 

10  9  8 

9  13  42 

9  24  51 

8  18  15 

8  29  25 

8  3  58 

8  15  8 


000 
054 
007 
041 


8  32 
19  41 
24  15 


5  24 

244 

9  58 

265 

21  7 

299 

14  32 

285 

25  41 

319 

0  15 

339 

11  24 

373 

4  49 

359 

15  58 

393 

VI 

000 
224 
330 
369 

^061  554 
:  095,  593 
081  738 
II5I778 
136  962 
, 170, 002 

'  1.56  147 
190  186 
210[371 
411 
595 
635 

780 
819 
004 
043 

188 
228 


VII  VIII  IX 


000 

875 
666 
694 

542 
570 
389 
417 
264 
293 

112 
140 
987 
015 
862 
891 

710 
738 
585 
613 
432 
461 


000 
045 
989 
023 

034 
068 
046 
080 
091 
124 

103 
136 
147 
182 
193 
227 

204 
238 

250 
283 
261 
295 


000 
111 
114 
150 

225 
261 
300 
336 
411 
447 

486 
522 
597 
633 
708 
744 

783 
819 


X 


000 
165 
313 
319 

478 
484 
638 
643 
802 
808 

962 
967 
126 
132 
291 
296 

451 
456 


894  I  615 
930  I  621 
969  775 
005  I  780 


XI  XII 


000 
290 
455 
496 

745 
787 
993 
034 
282 
324 

531 
572 
820 
862 
110 
152 

358 
400 
648 
690 
896 
938 


000 
043 
984 
018 

027. 
061 
036 
070 
079 
113 

088 
122 
131 
164 
173 
207 

182 
216 
225 
259 
234 
268 


54 


TABLE  XXXVII. 


Moori's  Motions  for  Days. 


D. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

Wo 

1 
13 

000 

1 

00000 

ouoo 

0000 

OUOO  i  0000 

0000 ; 0000 

0000 

0000 

000 

000 

2 

00274 

0650 

1040 

0287 : 0336 

0372  0058 

0390 

0024 

070 

031 

070 

034 

3 

00548 

1300 

2080 

0574 ' 0671 

0744 

OJ 15  0781 

0049 

140 

062 

141 

068 

4 

00S21 

1950 

3121 

0861  1007 

1116 

0173 

1171 

0073 

210 

093 

211 

103 

5 

01095 

2600 

4161 

1148  1342 

1488 

0231 

1561 

0097 

281 

125 

282 

137 

6 

01369 

3249 

5201 

1435  1678 

1860 

0289 

1952 

0121 

351 

156 

352 

171 

7 

0lf)43 

3899 

0241 

1722  2013 

2232 

0346 

2342 

0146  421 

187 

423 

205 

8 

0191(5 

4549 

7281 

2009  {  2349 

2604 

0404 

2732 

0170  491 

218 

493 

239 

9 

02190 

5199 

8321 

2296  2684 

2976 

0462 

3122 

0194  561 

249 

564 

273 

10 

02464 

5849 

9362 

2583 

3020 

3348 

0519 

3513 

0219  631 

280 

634 

308 

11 

02738 

6499 

0402 

2870 

3355 

3720 

0577 

3903 

0243  702 

311 

705 

342 

12 

03012 

7149 

1442 

3157 

3691 

4093 

0635 

4293 

0267  772 

342 

775 

376 

13 

032S5 

7799 

2482 

3444 

4026 

4465 ! 0692 

4684 

0291  842 

374 

845 

410 

14 

03559 

8449 

3522 

3731 

4362 

4837 

0750 

5074 

0316  912 

405 

916 

444 

15 

03833 

9098 

4563 

4018 

4698 

5209 

0SC8 

5404 

0340  982 

436 

986 

478 

10 

04107 

9748 

5603 

4305 

5033 

5581 

0866 

5854 

0364  052 

467 

057 

513 

17 

04380 

0308 

0643 

4592 

5369 

5953 

0923 

6245 

0389  122 

498 

127 

547 

18 

04654 

1048 

7683 

4878 

5704 

6325 

0981 

6635 

0413 

193 

529 

198 

581 

19 

0492S 

1698 

8723 

5165 

6040 

6697  1039 

7025 

0437 

263 

560 

268 

315 

20 

05202 

2348 

9763 

5452 

6375 

7069  1096 

7416 

0461 

333 

591 

339 

649 

21 

0547C 

2998 

0804 

5739 

6711 

7441  1154 

7806 

0486 

403 

623 

409 

683 

22 

05749 

3648 

1844 

6026 

7046 

7813  1212 

8196 

0510 

473 

654 

480 

718 

23 

06023 

4298 

2884 

6313 

7382 

8185  1269 

8586 

0534 

543 

685 

550 

752 

24 

06297 

4947 

3924 

6600 , 7717 

8557 ; 1327 

8977 

0559 

614 

716 

621 

786 

25 

06571 

5597 

4964 

6887  8053 

8929  1385 

9367 

0583 

684 

747 

691 

820 

26 

06844 

6247 

6005 

7174! 8389 

9301  1443 

9757 

0607 

754 

778 

762 

854 

27 

07118 

6897 

7045 

7401 i8724 

9673  1500 

0148 

0631 

824 

809 

832 

888 

28 

07392 

7547  8085 

7748  90C0 

0045  1558 

0538 

0656 

894 

840 

903 

923 

29 

07666 

8197  9125  [8035  9395 

0417' 1616 

0928 

0680 

964  872 

973 

957 

30 

07940 

8847  0165,8322  9731 

0789  1673 

1319 

0704 

034  903 

043 

991 

31 

08213 

9497 1 1205  18609  0066 

1161  1731 

1709 

0729 

105 : 934 

114 

025 

TABLE  XXXVII 


55 


Moon's  Motion  for  Days. 


D 

14 

15 

16 

17 

18 

19 

20 

[31 

22  23 

24 

25 

26 

27 

28 

29 

30 

31 

1 

000 

000 

000 

000 

000 

000 

000 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

2 

099 

031 

037 

042 

026 

002 

000 

01 

01 

10 

03 

04 

04 

07 

04 

03 

00 

00 

3 

198 

061 

073 

084 

052 

004 

001 

02 

02 

20 

05 

08 

07 

14 

OS 

06 

00 

00 

4 

297 

092 

110 

126 

078 

006 

001 

02 

03 

30 

OS 

12 

11 

21 

13 

09 

01 

01 

5 

397 

122 

146 

168 

104 

008 

002 

03 

03 

41 

11 

16 

15 

28 

17 

12 

01 

01 

6 

496 

153 

183 

210 

130 

Oil 

002 

04 

04 

51 

13 

21 

18 

35 

21 

15 

01 

01 

7 

595 

183 

220 

252 

156 

013 

003 

05 

05 

61 

16 

25 

22 

42 

25 

18 

01 

01 

8 

694 

214 

256 

294 

182 

015 

003 

05 

06 

71 

19 

29 

26 

49 

29 

22 

01 

02 

9 

793 

244 

293 

336 

208 

017 

004 

06 

07 

81 

21 

33 

30 

56 

33 

25 

01 

02 

10 

892 

275 

329 

379 

234 

019 

004 

07 

08 

91 

24 

37 

33 

63 

38 

28 

02 

02 

11 

992 

305 

366 

421 

260 

021 

005 

08 

09 

01 

27 

41 

37 

70 

42 

31 

02 

02 

12 

091 

336 

403 

463 

286 

023 

005 

08 

09 

11 

29 

45 

41 

77 

46 

34 

02 

03 

13 

190 

366 

439 

505 

312 

025 

005 

09 

10 

22 

32 

49 

44 

84 

50 

37 

02 

03 

14 

289 

397 

476 

547 

337 

028 

006 

10 

11 

32 

34 

53 

48 

91 

54 

40 

02 

03 

15 

388 

427 

512 

589 

363 

030 

006 

11 

12 

42 

37 

58 

52 

98 

58 

43 

02 

03 

16 

487 

458 

549 

631 

389 

032 

007 

11 

13 

52 

40 

62 

55 

05 

63 

46 

03 

04 

17 

587 

488 

586 

673 

415 

034 

007 

12 

14 

62 

42 

66 

59 

12 

67 

49 

03 

04 

18 

686 

519 

622 

715  441 

036 

008 

13 

14 

72 

45 

70 

63 

19 

71 

52 

03 

04 

19 

785 

549 

659 

757  467 

038 

008 

14 

15 

82 

48 

74 

66 

26 

75 

55 

03 

04 

20 

884 

580 

695 

799 1 493 

040 

009 

14 

16 

92 

50 

78 

70 

33 

79 

59 

03 

05 

21 

983 

611 

732 

841 '519 

042 

009 

15 

17 

03 

53 

S2 

74 

40 

84 

62 

03 

05 

22 

082 

641 

769 

883  545 

044' 010  1 

16 

18 

13 

56 

86 

77 

47 

88 

65 

04 

05 

23 

182 

672 

805 

925  571 

047  010 1 

17 

19 

23 

58 

90 

81 

54 

92 

68 

04 

05 

24 

281 

702 

842 

967  597 

049 

Oil 

17 

20 

33 

61 

95 

85 

61 

96 

71 

04 

06 

25 

380 

733 

878 

009  623 

051 

Oil 

18 

20 

43 

64 

99 

89 

68 

00 

74 

04 

06 

26 

479 

763 

915 

052  649 

053 

Oil 

19 

21 

53 

66 

03 

92 

75 

04 

77 

04 

06 

27 

578 

794 

952 

094  i 

675 

055 

012 

20 

22 

63 

69 

07 

96 

82 

09 

80 

04 

06 

28 

677 

824 

988 

136 

701 

057 

012 

20 

23 

73 

72 

11 

00 

89 

13 

83 

05 

06 

29 

777 

855 

025 

178 

727 

059 

013 

21 

24 

84 

74 

15 

03 

96 

17 

86 

05 

07 

30 

876 

885 

061 

220 

753 

061 

013 

22 

25 

94 

77 

19 

07 

03 

21 

89 

05 

07 

31 

975 

916 

098 

262 

779 

064' 014  1 

23 

26 

04 

80 

23 

11 

10 

25 

92  05 

07 

66 


TABLE  XXXVII. 


Maoris  Motions  for  Days. 


D. 

Evection. 

Anomaly. 

Variation. 

M.  Longitude. 

1 
2 
3 
4 
5 

i  °     ' 
0  0  0  0 
0  11  18  59 

0  22  37  59 

1  3  56  58 
1  15  15  58 

s 
0 
0 
0 

1 
1 

O        '        " 

0  0  00 
13  3  54.0 
26  7  47.9 

9  11  41.9 
22  15  35.9 

s  °  '  " 
0  0  0  0 
0  12  11  27 

0  24  22  53 

1  6  34  20 
1  18  45  47 

0  0  0  00 
0  13  10  35.0 

0  26  21  10.1 

1  9  31  45.1 
1  23  42  20.1 

6 
7 
8 
9 
10 

1  26  34  57 

2  7  53  57 

2  19  12  56 

3  0  31  55 
3  11  50  55 

2  5  19  29.8 

2  18  23  23.8 

3  1  27  17.8 
3  14  31  11.7 
3  27  35  5.7 

2  0  57  13 
2  13  8  40 

2  25  20  7 

3  7  31  34 
3  19  43  0 

2  5  52  55.1 

2  19  3  30.2 

3  2  14  5.2 
3  15  24  40.2 
3  28  35  15.2 

11 
12 
13 
14 
15 

3  23  9  54 

4  4  28  54 
4  15  47  53 
4  27  6  53 
6  8  25  52 

4 
4 
5 
5 
6 

10  38  59.7 
23  42  53.7 

6  46  47.6 
19  50  41.0 

2  54  35.6 

4  1  54  27 
4  14  5  54 

4  26  17  20 

5  8  28  47 
5  20  40  14 

4  11  45  50.3 

4  24  56  25.3 

5  8  7  0.3 

5  21  17  35.4 

6  4  28  10.4 

16 

17 
18 
19 
20 

5  19  44  51 

6  1  3  51 
6  12  22  50 

6  23  41  50 

7  5  0  49 

6 
6 
7 
7 
8 

15  58  29.5 
29  2  23.5 
12  6  17.5 
25  10  11.4 
8  14  5.4 

6  2  51  40 

6  15  3  7 
0  27  14  34 

7  9  26  1 
7  21  37  27 

6  17  38  45.4 

7  0  49  20.4 
7  13  59  55.5 

7  27  10  30.5 

8  10  21  5.5 

21 
22 
23 
24 
25 

7  16  19  49 

7  27  38  48 

8  8  57  47 

8  20  16  47 

9  1  35  46 

8 

9 

9 

10 

10 

21  17  59.4 
4  21  53.4 

17  25  47.3 
0  29  41.3 

13  33  35.3 

8  3  48  54 
8  16  0  21 

8  28  11  47 

9  10  23  14 
9  22  34  41 

8  23  31  40.5 

9  6  42  15.6 
9  19  52  50.6 

10  3  3  25.6 
10  16  14  0.7 

26 
27 
28 
29 
30 
31 

9  12  54  46 

9  24  13  45 

10  5  32  45 

10  16  51  44 

10  28  10  43 

11  9  29  43 

10  26  37  29.2 

11  9  41  23.2 
11  22  45  17.2 

0  5  49  11.1 

0  18  53  5.1 

1  1  56  59.1 

10  4  46  7 
10  16  57  34 

10  29  9  1 

11  11  20  28 
11  23  31  54 

0  5  43  21 

10  29  24  35.7 

11  12  35  10.7 
11  25  45  45.7 

0  8  56  20.8 

0  22  6  55.8 

1  5  17  30.8 

TABLE.   XXXVII. 


57 


Moon's  Motions  foi-  Days. 


D. 

Supp.  of  Node. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

1 

s  °       '     " 
0  0  0  0.0 

s   °   ' 
0  0  0 

000 

000 

000 

000 

000 

000 

000 

000 

2 

0  0  3  10.6 

0  11  9 

034 

039 

028 

034 

036 

005 

042 

034 

3 

0  0  6  21.3 

0  22  18 

068 

079 

056 

067 

072 

Oil 

083 

067 

4 

0  0  9  31.9 

1  3  27 

102 

118 

085 

101 

108 

016 

125 

101 

5 

0  0  12  42.5 

1  14  37 

136 

158 

113 

135 

143 

021 

166 

135 

6 

0  0  15  53.2 

1  25  46 

170 

197 

141 

169 

179 

027 

208 

168 

7 

0  0  19  3.8 

2  6  55 

204 

237 

169 

202 

215 

032 

250 

202 

8 

0  0  22  14.5 

2  18  4 

238 

276 

198 

236 

251 

037 

291 

235 

9 

0  0  25  25.1 

2  29  13 

272 

316 

226 

270 

287 

043 

333 

269 

10 

0  0  28  35.7 

3  10  22 

306 

355 

254 

303 

323 

048 

374 

303 

11 

0  0  31  46.4 

3  21  31 

340 

395 

282 

337 

358 

053 

416 

336 

12 

0  0  34  57.0 

4  2  40 

374 

434 

311 

371 

394 

058 

458 

370 

13 

0  0  38  7.6 

4  13  50 

408 

474 

339 

405 

430 

064 

499 

404 

14 

0  0  41  18.3 

4  24  59 

442 

513 

367 

438 

466 

069 

541 

437 

15 

0  0  44  28.9 

5  6  8 

476 

553 

395 

472 

502 

074 

583 

471 

16 

0  0  47  39.5 

5  17  17 

510 

592 

424 

506 

538 

080 

624 

505 

17 

0  0  50  50.2 

5  28  26 

544 

632 

452 

539 

573 

085 

666 

538 

18 

0  0  54  0.8 

6  9  35 

578 

671 

480 

573 

609 

090 

707 

572 

19 

0  0  57  11.5 

6  20  44 

612 

711 

508 

607 

645 

096 

749 

605 

20 

0  1  0  22.1 

7  1  53 

646 

750 

537 

641 

681 

101 

791 

639 

21 

0  1  3  32.7 

7  13  3 

680 

790 

565 

674 

717 

106 

832 

673 

22 

0  1  6  43.4 

7  24  12 

714 

829 

593 

708 

753 

112 

874 

706 

23 

0  1  9  54.0 

8  5  21 

748 

869 

621 

742 

788 

117 

915 

740 

24 

0  1  13  4.6 

8  16  30 

782 

908 

650 

775 

824 

122 

957 

774 

25 

0  1  16  15.3 

8  27  39 

816 

948 

678 

809 

860 

128 

999 

807 

26 

0  1  19  25.9 

9  8  48 

850 

987 

706 

843 

896 

133 

040 

841 

27 

0  1  22  36.5 

9  19  57 

884 

027 

734 

877 

932 

138 

082 

875 

28 

0  1  25  47.2 

10  1  6 

918 

066 

762 

910 

968 

143 

123 

908 

29 

0  1  28  57.8 

10  12  16 

952 

106 

791 

944 

003 

149 

165 

942 

30 

0  1  32  8.5 

10  23  25 

986 

145 

819 

978 

039 

154 

207 

975 

31 

0  1  35  19.1 

11  4  34 

020 

185 

847 

Oil 

075 

159 

248 

009  \ 

H 


98 


TABLE  XXXVIII. 

MooTi's  Motions  for  Hours. 


H. 

1 

2 

3 

4 

5 

6 

7' 

8 

9 

10 

11 

12 

13 

1 

1 

I    27 

43 

12 

14 

16 

2 

16 

1 

3 

1 

3 

1 

2 

2 

B    54 

87 

24 

28 

31 

5 

33 

2 

6 

3 

6 

3 

3 

3 

i        81 

130 

36 

42 

47 

7 

49 

3 

9 

4 

9 

4 

4 

4 

5   108 

173 

48 

56 

62 

10 

65 

4 

12 

5 

12 

6 

5 

5 

7      135 

217 

60 

70 

78 

12 

81 

5 

15 

6 

15 

7 

6 

6! 

?   162 

260 

72 

84 

93 

14 

98 

6 

18 

8 

18 

9 

7 

8 

)   190 

303 

84 

98 

109 

17 

114 

7 

20 

9 

20 

10 

8 

9 

I   217 

347 

96 

112 

124 

19 

130 

8 

23 

10 

23 

11 

9 

10 

3   244 

390 

108 

126 

140 

22 

146 

9 

26 

12 

26 

13 

10 

11' 

t   271 

433 

120 

140 

155 

24 

163 

10 

29 

13 

29 

14 

11 

12 

5   298 

477 

131 

154 

171 

26 

179 

11 

32 

14 

32 

16 

12 

13 

r   325 

520 

143 

168 

186 

29 

195 

12 

35 

16 

35 

17 

13 

14 

3   352 

563 

155 

182 

202 

31 

211 

13 

38 

17 

38 

18 

14 

16 

3   379 

607 

167 

196 

217 

34 

228 

14 

41 

18 

41 

20 

15 

17 

1   406 

650 

179 

210 

233 

36 

244 

15 

44 

19 

44 

21 

16 

18 

2   433 

693 

191 

224 

248 

38 

2G0 

16 

47 

21 

47 

23 

17 

19 

t   460 

737 

203 

238 

264 

41 

276 

17 

50 

22 

50 

24 

18 

20 

5   487 

780 

215 

252 

279 

43 

293 

18 

53 

23 

53 

25 

19 

21 

7   515 

823 

227 

266 

295 

46 

309 

19 

56 

25 

56 

27 

20 

22 

3   542 

867 

239 

280 

310 

48 

325 

20 

58 

26 

58 

28 

21 

23 

9   569 

910 

251 

294 

326 

50 

341 

21 

61 

27 

61 

30 

22 

25 

1   596 

953 

263 

308 

341 

53 

358 

22 

64 

28 

64 

31 

23 

26 

2   623 

997 

275 

322 

3.57 

55 

374 

23 

67 

30 

67 

33 

24 

27 

4   650 

1040 

287 

336 

372 

58  1  390 

24 

70  1  31 

70 

34 

Hours. 

Evection. 

Anomaly. 

Variation.   Longitude. 

- 

1 

O    '    " 

0  28  17 

O    '    " 

0  32  39.7 

o   /   "    o   '   " 
0  30  29   0  32  56.5 

2 

0  56  35 

1   5  19.5 

1  0  57   15  52.9 

3 

1  24  52 

1  37  59.2 

1  31  26   1  38  49  4 

4 

1  53  10 

2  10  39.0 

2  1  54 

2  11  45.8 

5 

2  21  27 

2  43  18.7 

2  32  23 

2  44  42.3 

6 

2  49  45 

3  15  58.5 

3  2  52 

3  17  38.8 

7 

3  18  2 

3  48  38.2 

3  33  20 

3  50  35.2 

8 

3  46  20 

4  21  18.0 

4  3  49 

4  23  31.7 

9 

4  14  37 

4  53  57.7 

4  34  17 

4  56  28.1 

10 

4  42  55 

5  26  37.5 

5  4  46 

5  29  24.6 

11 

5  11  12 

5  59  17.2 

5  35  15 

6  2  21.0 

12 

5  39  30 

6  31  57.0 

6  5  43 

6  35  17.5 

13 

6  7  47 

7  4  36.7 

6  36  12 

7  8  14.0 

14 

6  36  5 

7  37  16.5 

7  6  40 

7  41  10.4 

15 

7  4  22 

8  9  56.2 

7  37  9 

8  14  6.9 

16 

7  32  40 

8  42  36.0 

8  7  38 

8  47  3.4 

17 

8  0  57 

9  15  15.7 

8  38  6 

9  19  59.8 

18 

8  29  15 

9  47  55.5 

9  8  35 

9  52  56.3 

19 

8  57  32 

10  20  35.2 

9  39  3 

10  25  52.7 

20 

9  25  50 

10  53  15.0 

10  9  32 

10  58  49.2 

21 

9  54  7 

11  25  54.7 

10  40  1 

11  31  45.6 

22 

10  22  24 

11  58  34.5 

11  10  29 

12  4  42.1 

23 

10  50  42 

12  31  14.2 

11  40  58 

12  37  38.6 

24 

11  18  59 

13  3  54.0 

12  11  27 

13  10  35.0 

TABLE.   XXXVIII. 


59 


Moon^s  Motions  for  Hours. 


H. 

14 
4 

15 
1 

16 
2 

17 

1 

2 

2 

8 

3 

3 

4 

3 

12 

4 

5 

5 

4 

16 

5 

6 

7 

5 

21 

6 

8 

9 

6 

25 

8 

9 

11 

7 

29 

9 

11 

12 

8 

33 

10 

12 

14 

9 

37 

U 

14 

16 

10 

41 

13 

15 

18 

11 

45 

14 

17 

19 

12 

49 

15 

18 

21 

13 

54 

16 

20 

23 

14 

58 

18 

21 

25 

15 

62 

19 

23 

26 

16 

66 

20 

25 

28 

17 

70 

21 

26 

30 

18 

74 

23 

28 

32 

19 

78 

24 

29 

33 

20 

83 

25 

31 

35 

21 

87 

26 

32 

37 

22 

91 

28 

34 

39 

23 

95 

29 

35 

40 

24 

99 

31 

37 

42 

17   18  i  19  20  21  22  23  24  25  26  27  28  29 


1 

2 
3 
4 

5 
6 

8 
9 

10 
11 
12 
13 

14 
15 
10 
17 

18 
19 
21 
22 

23 

24 
25 
26 


0  0 

1  0 

1  0 

2  0 


H.  Sup.  of  Nod. 


0  7.9 

0  15.9 

0  23.8 

0  31.8 


39.7 
47.7 
55.6 
3.6 
11.5 
19.4 
27.4 
35.3 


0 
0 
0 
1 

1 
1 
1 
1 

1  43.3 
1  51.2 

1  59.2 

2  7.1 

2  15.0 

2  23.0 

2  30.9 

2  38.9 

2  46.8 

2  54.8 

3  2.7 
3  10.6 


II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

2 

XII 

1 

O    ' 

0  28 

1 

2 

1 

1 

1 

0 

0  56 

3 

3 

2 

3 

3 

0 

3 

3 

1  24 

4 

5 

4 

4 

4 

1 

5 

4 

1  52 

6 

7 

5 

6 

6 

1 

7 

6 

2  19 

7 

8 

6 

7 

7 

1 

9 

7 

2  47 

9 

10 

7 

9 

9 

1 

10 

9 

3  15 

10 

12 

8 

10 

10 

2 

12 

10 

3  43 

11 

13 

9 

11 

12 

2 

14 

11 

4  11 

13 

15 

11 

13 

13 

2 

15 

13 

4  39 

14 

16 

12 

14 

15 

2 

17 

14 

5   7 

16 

18 

13 

15 

16 

2 

19 

15 

5  35 

17 

20 

14 

17 

18 

3 

21 

17 

6   2 

18 

21 

15 

18 

19 

3 

23 

18 

6  30 

20 

23 

16 

19 

21 

3 

24 

19 

6  58 

21 

25 

18 

21 

22 

3 

26 

21 

7  26 

23 

26 

19 

22 

24 

4 

28 

22 

7  54 

24 

28 

20 

24 

25 

4 

29 

24 

8  22 

26 

29 

21 

25 

27 

4 

31 

25 

8  50 

27 

31 

22 

27 

28 

4 

33 

27 

9  18 

28 

32 

24 

28 

30 

4 

35 

28 

9  45 

30 

34 

25 

29 

31 

5 

37 

29 

10  13 

31 

36 

26 

31 

33 

5 

38 

31 

10  41 

33 

38 

27 

32 

34 

5 

40 

32 

11   9 

34 

39 

28 

34 

36 

5 

42 

34 

60 


TABLE  XXXIX. 


Moon's  Motions  for  Minutes. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

1 

0 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2 

0 

1 

1 

0 

0 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

3 

1 

2 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

4 

2 

3 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

5 

1 

2 

4 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

6 

3 

4 

1 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

7 

3 

5 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

8 

2 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

9 

2 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

10 

2 

5 

7 

2 

2 

3 

0 

3 

0 

0 

0 

0 

0 

1 

0 

0 

0 

0 

11 

2 

5 

8 

2 

3 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

12 

2 

5 

9 

2 

3 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

13 

2 

6 

9 

3 

3 

3 

4 

0 

0 

0 

0 

0 

0 

0 

14 

3 

6 

10 

3 

3 

4 

4 

0 

0 

0 

0 

0 

0 

0 

15 

3 

7 

11 

3 

3 

4 

4 

0 

0 

0 

0 

0 

0 

0 

16 

3 

7 

12 

3 

4 

4 

4 

0 

0 

0 

0 

0 

0 

0 

17 

3 

8 

12 

3 

4 

4 

5 

0 

0 

0 

0 

0 

0 

0 

18 

3 

8 

13 

4 

4 

5 

5 

0 

0 

0 

0 

0 

0 

19 

4 

9 

14 

4 

4 

5 

5 

0 

0 

0 

0 

0 

0 

20 

4 

9 

14 

4 

5 

5 

5 

0 

0 

0 

0 

0 

21 

4 

10 

15 

4 

5 

5 

6 

0 

0 

0 

1 

0 

1 

0 

22 

4 

10 

16 

4 

5 

6 

6 

0 

0 

2 

0 

0 

23 

4 

10 

17 

5 

5 

6 

6 

0 

0 

2 

0 

0 

24 

5 

11 

17 

5 

6 

6 

7 

0 

2 

0 

25 

5 

11 

18 

5 

6 

6 

7 

0 

2 

0 

26 

5 

12 

19 

5 

6 

7 

7 

0 

2 

0 

27 

5 

12 

19 

5 

6 

7 

7 

0 

2 

0 

28 

5 

13 

20 

6 

7 

7 

8 

0 

2 

0 

29 

6 

13 

21 

6 

7 

7 

8 

0 

2 

0 

30 

6 

14 

22 

6 

7 

8 

8 

0 

2 

1 

0 

TABLE  XXXIX. 


61 


Moon's  Motions  for  Minutes. 


Min. 

Evec. 

Anom. 

Varia. 

Long. 

Sup. 
Nod. 

II 

V 

VI  VII 

vm 

IX 

XI 

XII 

1 

0  28 

0  32.7 

0  30 

0  32.9 

0.1 

0 

0 

0 

0 

0 

0 

0 

0 

2 

0  57 

1  5.3 

1  1 

1  5.9 

0.3 

1 

0 

0 

0 

0 

0 

0 

0 

3 

1  25 

1  38.0 

1  31 

1  38.8 

0.4 

1 

0 

0 

0 

0 

0 

0 

0  1 

4 

1  53 

2  10.6 

2  2 

2  11.8 

0.5 

2 

0 

0 

0 

0 

0 

0 

0 

5 

2  2) 

2  43.3 

2  32 

2  44.7 

0.7 

2 

0 

0 

0 

0 

0 

0 

0 

6 

2  50 

3  16.0 

3  3 

3  17.6 

0.8 

3 

0 

0 

0 

0 

0 

0 

0 

7 

3  18 

3  48.6 

3  33 

3  50.6 

0.9 

3 

0 

0 

0 

0 

0 

0 

0 

8 

3  46 

4  21.3 

4  4 

4  23.5 

1.1 

4 

0 

0 

0 

0 

0 

0 

0 

9 

4  15 

4  54.0 

4  34 

4  56.5 

1.2 

4 

0 

0 

0 

0 

0 

0 

0 

10 

4  43 

5  26.6 

5  5 

5  29.4 

1.3 

5 

0 

0 

0 

0 

0 

0 

0 

11 

5  11 

5  59.3 

5  35 

6  2.4 

1.5 

5 

0 

0 

0 

0 

0 

0 

0 

12 

5  40 

6  31.9 

6  6 

6  35.3 

1.6 

6 

0 

0 

0 

0 

0 

0 

0 

13 

6  8 

7  4.6 

6  36 

7  8.2 

1.7 

6 

0 

0 

0 

0 

0 

0 

0 

14 

6  36 

7  37.3 

7  7 

7  41.2 

1.9 

7 

0 

0 

0 

0 

0 

0 

0 

15 

7  4 

8  9.9 

7  37 

8  14.1 

2.0 

7 

0 

0 

0 

0 

0 

0 

0 

16 

7  33 

8  42.6 

8  8 

8  47.1 

2.1 

7 

0 

0 

0 

0 

0 

1  0 

0 

17 

8  1 

9  15.3 

8  38 

9  20.0 

2.3 

8 

0 

0 

0 

0 

0 

^  0 

0 

18 

8  29 

9  47.9 

9  9 

9  52.9 

2.4 

8 

0 

0 

0 

0 

0 

0 

19 

8  58 

10  20.6 

9  39 

10  25.9 

2.5 

9 

0 

0 

0 

0 

0 

0 

20 

9  26 

10  53.2 

10  10 

10  58.8 

2.6 

9 

0 

1 

0 

0 

0 

0 

21 

9  54 

11  25.9 

10  40 

11  31.8 

2.8 

10 

0 

0 

0 

0 

0 

22 

10  22 

11  58.6 

11  11 

12  4.7 

2.9 

10 

0 

0 

0 

23 

10  51 

12  31.2 

11  41 

12  37.6 

3.0 

11 

0 

0 

0 

24 

11  19 

13  3.9 

12  12 

13  10.6 

3.2 

11 

0 

25 

11  47 

13  36.6 

12  42 

13  43.5 

3.3 

12 

0 

26 

12  16 

14  9.2 

13  13 

14  16.5 

3.4 

12 

27 

13  44 

14  41.9 

13  43 

14  49.4 

3.6 

13 

28 

13  12 

15  14.6 

14  13 

15  22.3 

3.7 

13 

29 

13  40 

15  47.2 

14  44 

15  55.3 

3.8 

13 

30 

14  9 

16  19.9 

15  14 

16  28.2 

4.0 

14 

TABLE  XXXIX. 


Moon's  Motions  for  Minutes. 


1  1   2 

3 

4   5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

31 

6  14 

22 

6 

7 

8 

8 

0 

1 

1 

2 

32 

6 

14 

23 

6 

7 

8 

9 

2 

1 

2 

2 

33 

6 

15 

24 

7 

8 

9 

9 

2 

2 

2 

34 

6 

15 

25 

7 

8 

9 

9 

2 

2 

2 

35 

7 

IG 

25 

7 

8 

9 

10 

2 

2 

2 

36 

7 

16 

25 

7 

8 

9 

10 

2 

2 

3 

37 

7 

17 

27 

7 

9 

10 

10 

2 

2 

3 

38 

7 

17 

27 

8 

9 

10 

2 

10 

2 

2 

3 

39 

7 

18 

23 

8 

9 

10 

2 

11 

2 

2 

3 

40 

8 

18 

29 

8 

9 

10 

2 

11 

2 

2 

3 

41 

8 

19 

30 

8 

10 

11 

2 

11 

2 

2 

3 

42 

8 

19 

30 

8 

10 

11 

2 

11 

2 

2 

3 

43 

8 

19 

31 

9 

10 

11 

0 

12 

1 

2 

2 

3 

44  8 

20 

32 

9 

10 

11 

2 

12 

2 

2 

3 

45  9 

20 

32 

9 

10 

12 

2 

12 

2 

2 

3 

46  9 

21 

33 

9 

11 

12 

2 

12 

2 

1 

2 

3 

47  9 

21 

34 

9 

11 

12 

2 

13 

2 

2 

3 

48  9 

22 

35 

10 

11 

12 

2 

13 

2 

2 

3 

49  9 

22 

35 

10 

11 

13 

2 

13 

2 

2 

3 

SO  9 

23 

36 

10 

11 

13 

2 

13 

2 

2 

3 

51  10 

23 

37 

10 

12 

13 

2 

14 

2 

2 

4 

52  10 

24 

38 

10 

12 

13 

2 

14 

3 

3 

4 

53  10 

24 

38 

11 

12 

14 

2 

14 

3 

3 

4 

54  10 

24 

39!  11 

12 

14 

2 

14 

3 

3 

4 

2 

55  10 

25 

40 

11 

13 

14 

2 

15 

3 

3 

4 

2 

56 

11 

25 

40 

11 

13 

14 

2 

15 

1 

3 

3 

4 

2 

57 

11 

26 

41 

11 

13 

15 

2 

15 

3 

3 

4 

2 

58 

11 

26 

42 

12 

13 

15 

2 

16 

3 

3 

4 

2 

2 

59  11 

27 

43  12 

14 

15 

2 

16 

3 

3 

1   4 

2 

2 

60|11 

27 

43  12 

14 

15 

2 

16 

3 

3 

I   4 

2 

2 

TABLE  XXXIX. 


63 


MoorCs  Motions  for  Minutes. 


Min. 

Evcc. 

Anom. 

Varia. 

Long. 

Sup. 
Nod. 

.1 

V 

VI 

VII 

vm 

IX 

XI 

XII 

31 

14  37 

16  52.5 

15  45 

17  1.2 

4.1 

14 

32 

15  5 

17  25.2 

16  15 

17  34.1 

4.2 

15 

33 

15  34 

17  57.9 

16  46 

IS  7.1 

4.4 

15 

34 

16  2 

18  30.5 

17  16 

18  40.0 

4.5 

16 

35 

16  30 

19  3.2 

17  47 

19  12.9 

4.7 

16 

36 

16  58 

19  35.8 

18  17 

19  45.9 

4.8 

17 

37 

17  27 

20  8.5 

18  48 

20  18.8 

4.9 

17 

38 

17  55 

20  41.2 

19  18 

20  51.8 

5.0 

18 

^ 

39 

18  23 

21  13.8 

19  49 

21  24.7 

5.2 

18 

40 

18  52 

21  46.5 

20  19 

21  57.6 

5.3 

19 

41 

19  20 

22  19.2 

20  50 

22  30.6 

5.4 

19 

42 

19  48 

22  51.8 

21  20 

23  3.5 

5.6 

20 

43 

20  16 

23  24.5 

21  51 

23  36.5 

5.7 

20 

i  ^ 

44 

20  45 

23  57.1 

22  21 

24  9.4 

5.8 

21 

1 

45 

21  13 

24  29.8 

22  52 

24  42.3 

6.0 

21 

1 

1 

46 

21  41 

25  2.5 

23  22 

25  15.3 

6.1 

21 

1 

47 

22  10 

25  35.1 

23  53 

25  48.2 

6.2 

22 

1 

48 

22  38 

26  7.8 

24  23 

26  21.2 

6.4 

22 

49 

23  6 

26  40.5 

24  54 

26  54.1 

6.5 

23 

50 

23  34 

27  13.1 

25  24 

27  27.0 

6.6 

23 

51 

24  3 

27  45.8 

25  55 

28  0.0 

6.8 

24 

1 

52 

24  31 

28  18.5 

26  25 

28  32.9 

6.9 

24 

53 

24  59 

28  51.1 

26  56 

29  5.9 

7.0 

25 

54 

25  28 

29  23.8 

27  26 

29  38.8 

7.1 

25 

2 

55 

25  56 

29  56.4 

27  56 

30  11.8 

7.3 

26 

2 

56 

26  24 

30  29.1 

28  27 

30  44.7 

7.4 

26 

2 

57 

26  52 

31   1.8 

28  57 

31  17.6 

7.5 

27 

2 

2 

58 

27  21 

31  34.4 

29  28 

31  50  6 

7.7 

27 

2 

2 

59 

27  49 

32  7.1 

29  58 

32  23.5 

7.8 

28 

2 

2 

60 

28  17 

32  39.8 

30  29 

32  56.5  7.9 

28 

2 

1    1 

2 

JJ 

64 


TABLE  XL. 


Moon^s  Motions  for  Seconds. 


Sec. 

Evec. 

Anom. 

Var. 

Long. 

Sec. 

Evec. 

Anom. 

Var. 

Long. 

1 

0 

0.5 

1 

0.5 

31 

15 

16.9 

16 

17.0 

2 

1 

1.1 

1 

1.1 

32 

15 

17.4 

16 

17.6 

3 

1 

1.6 

2 

1.0 

33 

16 

18.0 

17 

18.1 

4 

2 

2.2 

2 

2.2 

34 

16 

18.5 

17 

18.7 

5 

2 

2.7 

3 

2.7 

35 

17 

19.1 

18 

19.2 

6 

3 

3.3 

3 

3.3 

36 

17 

19.6 

18 

19.8 

7 

3 

3.8 

4 

3.8 

37 

18 

20.1 

19 

20.3 

8 

4 

4.3 

4 

4.4 

38 

18 

20.7 

19 

20.9 

9 

4 

4.9 

5 

4.9 

39 

18 

21.2 

20 

21.4 

10 

6 

6.4 

5 

5.5 

40 

19 

21.8 

20 

22.0 

11 

5 

6.0 

6 

6.0 

41 

19 

22.3 

21 

22.5 

12 

6 

6.5 

6 

6.6 

42 

20 

22.9 

21 

23.1 

13 

6 

7.1 

7 

7.1 

43 

20 

23.4 

22 

23.6 

14 

7 

7.6 

7 

7.7 

44 

21 

24.0 

22 

24.2 

15 

7 

8.2 

8 

8.2 

45 

21 

24.5 

23 

24.7 

16 

8 

8.7 

8 

8.8 

46 

22 

25.0 

23 

25.3 

17 

8 

9.2 

9 

9.3 

47 

22 

25.6 

24 

25.8 

18 

9 

9.8 

9 

9.9 

48 

23 

26.1 

24 

26.4 

19 

9 

10.3 

10 

10.4 

49 

23 

26.7 

25 

26.9 

20 

9 

10.9 

10 

11.0 

50 

24 

27.2 

25 

27.4 

21 

10 

11.4 

11 

11.5 

51 

24 

27.8 

26 

28.0 

22 

10 

12.0 

11 

12.1 

52 

25 

28.3 

20 

28.5 

23 

11 

12.5 

12 

12.6 

53 

25 

28.9 

27 

29.1 

24 

11 

13.1 

12 

13.2 

54 

26 

29.4 

27 

29.6 

25 

12 

13.6 

13 

137 

55 

26 

29.9 

28 

30.2 

26 

12 

14.1 

13 

14.3 

56 

26 

30.5 

28 

30.7 

27 

13 

14.7 

14 

14.8 

57 

27 

31.0 

29 

31.3 

28 

13 

15.2 

14 

15  4 

58 

27 

31.6 

29 

31.8 

29 

14 

15.8 

15 

15.9 

59 

28 

32.1 

30 

324 

30 

14 

16.3 

15 

16.5 

60 

28 

32.7 

30 

32.9 

TABLE  XLI.  65 

First  Equation  of  Maori's  Longitude. — Argument  1. 


Arg. 

0 
50 
100 
150 
300 
250 

300 
350 
400 
450 
500 

550 
600 
650 
700 
750 

800 
850 
900 
950 
1000 

1050 
1100 
1150 
1200 
1250 

1300 
1350 
1400 
1450 
1500 


Diff. 
for  10 


1550    3 
1600    3 


1650 
1700 
1750 

1800 
1850 
1900 
1950 
2000 

2050 
2100 
2150 
2200 
2250 

2300 
2350 
2400 
2450 
2500 


40.0 

18.8 
57.7 
36.6 
15.6 
54.7 

33.9 
13.2 
52.6 
32.3 
12.1 

52.1 
32.4 
13.0 
53.8 
34.9 

16.4 

58.2 

40.3 

22.8 

5.7 

49.0 
32.8 
17.0 
1.6 
46.7 

32.3 

18.4 

5.0 

52.2 

39.9 

28.1 
16.9 
63 
56.3 
46.8 

38.0 
29.7 
22.1 
15.1 
8.8 

3.1 
58.0 
53.6 
49.8 
46.7 

44.2 
42.3 
41.1 
40.6 
40.7 


4.24 
4.22 
4.22 
4.20 

4.18 1; 

4.16 


A  Iff. 


2500 
2550 
2600 
2650 
2700 
2750 


4.14 
4.12 
4.06 
4.04 
4.00 

3.94 

3.88 
3.84 
3.78 
3.70 

3  64 
3.5.S 
3.50 
3.42 
3.34 


324 
3.16 
3.08 

2.98 
2.88 

2.78 
2.68 
2.56 
2.46 
2.36 

2.24 
2.12 
2.00 
1.90 
1.76 

1.66 
1.52 
1.40 
1.26 
1.14 

1.02 
0.88 
0.76 
0.62 
0.50 


2800 
2850 
2900 
2950 
3000 

3050 
3100 
3150 
3200 
3250 

3300 
3350 
3400 
3450 
3500 

3550 
3600 
3650 
3700 
"7.50 

3800 
3850 
3900 
3950 
4000 


4050 
4100 
4150 
4200 
4250 

4300 
4350 
4400 
4450 
4500 


0.38 
0.24 
0.10 


0.02 


1  40.7 

1  41.5 

i  42.9 

1  45.0 

1  47.7 

1  51.0 

1  .55.0 

1  59.6 

2  4.8 
2  10.7 
2  17.1 


2  24.2 

2  31.9 

2  40.1 

2  48.9 

2  58.3 

3  8.2 
3  18.7 
3  29.7 
3  41.3 
3  53.4 


Dift: 
for  10 


5.9 
19.0 
32.5 
46.5 

0.9 

15.8 
31.0 
46.7 
2.8 
19.2 

36.0 
53.1 
10.6 

28.4 
40.4 


8     4.7 

8  23.3 

8  42 

9  1.2 
9  20.4 

9  39.9 

9  59.5 

10  19.2 

10  39.1 

10  59.1 


4550 
4600 
4650 
4700 
4750 

4800 
4850 
4900 
4950 
5OOOI12  40.0 


11 

il2 


19.1 
39.3 
59.5 
19.7 


0.16 
0.28 
0.42 
0.54 
0.66 
0.80 

0.92 
1.04 
1.18 
1.28 
1.42 

1.54 
1.64 
1.76 
1.88 
1.98 

2.10 
2.20 
2.32 
2.42 
2.50 

2.62 
2.70 
2.80 
2.88 
2.98 

3.04 
3.14 
3.22 
3.28 
3.36 

3.42 
3.50 
3.56 
3.60 
3.66 

3.72 

3.78 
3.80 
3.84 
3.90 

3.92 
3.94 
3.98 
4.00 
4.00 

4.04 
4.04 
4.04 
4.06 


Are 


5000 
5050 
5100 
51.50 
5200 
5250 

5300 
5350 
5400 
5450 
5500 

5.550 
5600 
5650 
5700 
5750 


5800  17 
.58.50  18 


Diff. 
for  10 


■i900 
5950 
6000 

6050 
6100 
6150 
6200 
6250 

6300 
6350 
6400 
6450 
6500 


6,550  21 
6600i21 
6650 1 22 
070022 
0750,22 

680022 
6850122 
6900  22 
0950  22 
7000!  23 

705023 
7100  23 
7150le3 
7200123 
7C50,23 

7300,23 
7350  23 
7400  23 
7450!  23 
75OOI23 


40.0 
0.3 

20.5 

40.7 
0.9 

20.9 

40.9 
0.8 
20.5 
40.1 
59.6 

18.8 
37.8 
56.7 
15.3 
33.6 

51.6 

9.4 

26.9 

44.0 

0.8 

17.2 
33.3 
49.0 
4.2 
19.1 

33.5 
47.5 
1.0 
14.1 
26.6 

38.7 
50.3 
1.3 
11.8 
21.7 

31.1 
39.9 

48.1 

55.8 

2.9 

9.3 
15.2 
20.4 
25.0 
29.0 

32.3 
35.0 
37.1 
38.5 
39.3 


4.06 
4.04 
4.04 
4.04 
4.00 
4.00 

3.98 
3.94 
3.92 
3.90 
3.84 

3.80 
3.78 
3.72 
3.66 
3.60 

3.56 
3.50 
3.42 
3.36 

3.28 

3.22 
3.14 
3.04 
2.98 

2.88 

2.80 
2.70 
2.62 
2.50 

2.42 

2.32 
2.20 
2.10 
1.98 
1.88 

1.76 
1.64 
1  54 
1.42 
1.28 

1.18 
1.04 
0.92 
0.80 
0.66 

0.54 
0.42 
0.28 
0.16 


Arg-. 


Diff. 
for  10 


7500  23 

755023 
7600  23 
7650  23 


7700 
7750 

7800 
7850 
7900 
7950 
8000 

8050 
8100 
8150 
8200 
8250 


23 
23 

23 
23 
23 
23 
23 

23 

22 
22 
22 
22 

22 


8300 
8350|22 
8400^22 
8450  21 
8500.21 

8550121 
860021 
8650  21 


8700 
8750 


8800  20 
8850  20 


8900 
8950 
9000 


90.50  IS 
9100  18 
91.50J18 
9200  18 
9250  17 


9300 
9350 
9400 
9450 


9500J16 

9550|l5 
9600  15 


9650 
9100 
9750 


9800  14 
9850  13 


9900 

9950 

10000 


39.3 
39.4 
88.9 
37.7 
35.8 
33.3 

30.2 
26.4 
22.0 
16.9 
11 

4.9 
57.9 
50.3 
42.0 
33.2 

23.7 
13.7 
3.1 
51.9 
40.1 

27.8 
15.0 
1.6 
47.7 
33.3 

18.4 
3.0 
47.2 
31.0 
H.3 

57.2 
39.7 
21.8 
3.6 
45.1 

26.2 

7.0 

47.6 

27.9 

7.9 

47.7 
27.4 
6.8 
46.1 
25.3 

4.4 
43.4 
22.3 

1.2 

4o.o 


0.02 


0.10 
0.24 
0.38 
0.50 
0.62 

0.76 
0,88 
1.02 
1.14 
1.26 

1.40 
1.52 
1.66 
1.76 
1.90 

2.00 
2.12 
2.24 
2.36 

2.46 

2.56 
2.68 

2.78 
2.88 
2.98 

3.08 
3  16 
3.24 
3.34 
3.42 

3.50 
3.58 
3.64 
3.70 
3.78 

3.84 
3.88 
3.94 
4.00 
4.04 

4.06 
4.12 
4.14 
4.16 
4.18 

4.26 
4.22 
4.22 
4.214 


I 


«6  TABLE  XLII. 

Equations  2  to  7  of  Moon's  Longitude.     Arguments  2  to  7. 


Arg. 


2500 
2600 
2700 
2800 
2900 
3000 

3100 
3200 
3300 
3400 
3500 

3600 
3700 
3800 
3900 
4000 

4100 
4200 
4300 
4400 
4500 

4600 
4700 
4800 
4900 
5000 


4  57.3 
4  57.0 
4  56.1 
4  54.7 
4  52.7 
4  50.1 


0.3 
0.9 

1.4 
2.0 
2.6 
3.1 
3.7 
4.2 
4.7 
5.2 
5.7 
6.1 
6.6 
6.9 
7.3 
7.7 
3  48.9  7  9 

3  41.0s:3 

3:«.7  8.5 

3  24.2|8.7 


4  47.0 
4  43.3 
4  39.1 
4  34.4 
4  29  2 

4  23  5 
417  4 
4  10.8 
4  3.9 
3  56.6 


ditf 


5100 
5200 
5300 
5400 
.5500 

15600 
5700 
5800 
'■5900 
16000 

6100 


3  15.5 

3    6.6 

2  57.6 
2  48.5 
2  39.3; 
2  30.0 


9.2 

9.3 
9.1 
9.0 

8.9 

8.7 

8.5 
S.3 
7.9 
7.7 
7.3 
6.9 

j6300'0  42.6:!!-^ 
6400  0  36.5°:, 

6500  0  30.81 

5.2 

6600;0  25.C!.  „ 
6700  0  20.9!*^ 
6800|0  16.7*~ 
6900  013.0'3^ 


2  20.8 
2  11.5 
2  2.4 
153.4 
144.5 

135.8 
1  27.3 
1  19.0 
1  11.1 
1    3.4 

056.1 


:0200  0  49.21 


7000  0    9.9 


7100:0 

7200  0 


7300 
7400 
7500 


7.3 
5.3 
3.9 
3.0 
2.7 


diff 


23 

2.4 
2.8 
3.3 
4.1 
5.1 

6.4 
7.8 
0  9.4 
011.3 
0  13.3 

0  15.5 
0  17.9 
0  20.5 
0  23.2 
0  26.1 

0  29.1 
0  32.2 
0  35.4 
0  3S.8 
0  42.2 

0  45.7 
0  49.2 

0  52.8 : 

0  56.4 
0.0 


3.6 


1  7.2 
1  10.8 
1  14.3 
1  17.Si 

121.2' 
124.6 
127.8 
1  30.9 
133.9 

136.8 
1  39.5 
142.1 
144  5 
146.7 

148.7! 
1  50.6 
1  52.2 
1536 
154  9 

1  55.9 
156.7 
157.2 
157.6 
157.7 


0.1 

10.4 
0.5 
!0.8 
1.0 
1.3 
1.4 
1.6 
1.9 
2.0 
2.2 
2.4 
2.6 
2.7 
2.9 
3.0 
3.1 
3.2 
3.4 
3.4 
3.5 
3.5 
3.6 
3.0 
J3.6 
3.6 
[3.6 
3.6 
3.5 
'3.5 
13.4 

3.4 
3.2 
3.1 
3.0 
!2.9 

2.7 
2.6 
2.4 

M.2 


diff 


12.0 
1.9 
1.6 
1.4 
1.3 
ll.O 

0.8 
0  5 
0.4 
0.1 


6  30.3 
6  29.9 

6  28.8 
6  26.9 
6  24.3 
621.0 

6  16.9 
6  12.2 
6  6.8 
6  0.7 
5  54.0 

5  46.6 
5  38.7 
5  30.3 
521.3 
5  11.9 


9.0 
9.4 
9.9 
10.3 
10.7 
10.9 
11.3 
11.5 
11.6 
11.8 
12.0 
11.9 
11.9 

3    Sl'12.0 
2  56.1' 


5    2.0 

451.7 
441.0 
4  30.1 
4  18.8 

4  7.3 
3  55.7 
3  43.9 
3  31.9 
3  20.0 


0.4 
1.1 
1.9 
2.6 
3.3 
4.1 
4.7 
5.4 
6.1 
6.7 
7.4 
7.9 


2  44.3{ 
2  32.7 
22I.2I 


11.8 
11.6 
11.5 


11.3 
2    9.9!i0.9 
159.0 
148.3 

(i 
128.1 


1  18.7 
1  9.7 
1  1.3 
0  53.4 
0  46.0 

0  39.3! 

0  33.2! 

0  27 £ 
0  23.1 
0  19.0 

0  15.7 
0  13.1 
011.2 
0  10.1 
0    97 


10.7 

10.3 

9.9 

9.4 

9.0 
8.4 
7.9 
7.4 
6.7 
6.1 
5.4 
4.7 
41 
3.3 

2.6 
1.9 
1.1 
0.4 


3  39.4 
3  39.2 
3  38.5 
3  37.5 
3  36.0 
3  34.1 

331.7 
3  29.0 
3  25.9 
3  22.4 
3  18.5 

3  14  3 

3  9.7 
3  4.9 
2  59.7 
2  54  3 

2  4S.r 
2  42.7 
2  36.fi 
■2  30.3 
2  23.8 

2  17.2 
2  10.5 
2  3.7 
156.9 
150.0 

143  1 
1  36.3 
129.5 
1  22.8 
1  16.2 

1  9.7 
1  3.4 
0  57.3 
051.4 
0  45.7 

0403 
0  35.1 
0  30.3 
0  25.7 
0  21.5 

0  17.6 
0  14.1 
011.0 
0  8.3 
5.9 


diff 


diff 


6.2 


0 
0 
0 
0 
0 
0  10.3 


6.4 
6.9 

7.7 


0.2 
0.5 
0.8 
1.1 
1.5 
[1.8 
012.li2i 
0  14.2^1 
0  16.6  7,  fi 

0  19  «>  ^ 
'  ^^ "  3  0 
0  22.2 

3.2 


0  25.4.: 
0  28.9 
0  32.7 


3.5 

3.8 

3  9 
0  36.6  4j 

0  40.71,' 

4.4 

0  45.1  45 
049.6  47 

0  54  3!4'9 
059.249 

1  ^-^15:1 
1  9251 
114  3'5  2 


1  19.5 

1  24.7 
130.0 


5.3 


135.3 

1  40.5 
145.7 
150.8 


5.2 

5.2 
5.1 

155.9;j 

2  08  49 
2  5.7:47 
210.44-5 

2i4.9:rj 

2193.- 

4.1 

223.4' 
■2  27.3  J; 
'231.1^-^ 
I2  34.6^0 

2  40.8' 
243.4:JJ 
2  45.8'^-* 
1247.9:' 
249.7|^-® 

1.5 
2  51  2' 

1  1 
2  52  3 

2  53.1 1  Jl 

2  53.6"^ 

2  53.8" 


diff  Arg. 


0.8 
0.9 
1.3 

1.8 
2.7 
3.7i 

5.01 
6.4 


0.1 
0.4 
0.5 
0.9 
1.0 
1.3 
1.4 
1.7 
1.9 


OIO.OL,  1 
0  12.1L-„ 


0  14.4 
0  16.8 


0  19.5  2.8 


0  22.3 
0  25.2 

0  28.3 
031  5 
0  34.8 
0  38.2 
041.7 

0  45.3 
0  48.9 
0  52.6 

0  56.3 

1  0.0 

1    3.7 

1  7.4 
1  11.1 
1  14  7 
1  18  3 

121.8 
125.2 
128.5 
131.7 
134.8 

137.7 
1  40.5 
143.2 
145.6 
147.9 

1  50.0 
I  51. 9| 
1  53.6 
1  55.0 
156  3 

1  57.3 
1  58.2 
1  58.7 
159.1 
159.2 


2.9 
3.1 
3.2 
3.3 
3.4 
3.5 
3.6 
3.6 
3.7 
3.7 
3.7 
3.7 

3.7 
3.7 
3.6 
3.6 
3.5 

3.4 
3.3 
3.2 
3.1 
2.9 

2.8 
2.7 
2.4 
2.3 
2.1 

1.9 
1.7 
1.4 
1.3 
1.0 

0.9 
05 
0.4 
0.1 


2500 
2400 
2300 
2200 
2100 
2000 

'l900 
1800 
1700 
1600 
1500 

1400 
1300 
1200 
1100 
1000 

900 
800 
700 
600 
500 

400 
300 
200 
100 
0 

9900 
9800 
9700 
9600 
9500 

9400 
9300 
9200 
9100 
9000 

8900 
8800 
8700 
8600 
8500 

8400 
8300 
8200 
8100 
8000 

7900 
7800 
7700 
7600 
7500 


TABLE  XLIII. 


TABLE  XLIV. 


67 


Equations  8  and  9. 


Equations   10  and  IL 


Arg. 
0 

8 

1  20  0 

Arg. 
5000 

8 
1  20.0 

1  20.0 

Arg. 
0 

10 

11 
10.0 

Arg. 
500 

10 
10.0 

11 
10.0 

1  20.0 

10.0 

100 

1  15.5 

1  28  7 

5100 

1  24.4 

1  25.8 

10 

9.3 

11. li 

510 

9.6 

10.8 

200 

1  11.1 

1  37.3 

5200 

1  28.8 

1  31.4 

20 

8.6 

12.1 

520 

9.2 

11.5 

300 

1  6.7 

1  45.7 

5300 

1  33.1 

1  36.9 

30 

8.0 

13.1' 

530 

8.9 

12.3 

400 

1  2.3 

1  53.7 

5400 

1  37.4 

1  42.0 

40 

7.4 

14.1 : 

540 

8.5 

12.9 

500 

0  58.0 

2  1.3 

5500 

1  41.6 

1  46.8 

50 

6.8 

15.0 

550 

8.2 

13.6 

600 

0  53.8 

2  8.3 

5600 

1  45.8 

1  51.0 

60 

6.2 

158 

560 

7.9 

14.2 

700 

0  49.7 

2  14.7 

5700 

1  49.8 

1  54.6 

70 

5.7 

16.6 

570 

7.7 

14.6 

800 

0  45.7 

2  20.2 

5800 

1  53.8 

1  57.6 

80 

53 

17.3 

580 

7.5 

15.0 

900 

0  41.9 

2  25.0 

5900 

1  57.6 

1  59.8 

90 

4.9 

17.9 

5U0 

7.4 

15.4 

1000 

0  38.2 

2  28.9 

6000 

2  1.2 

2  1.3 

100 

4.6 

18.3' 

600 

7.3 

15.6 

1100 

0  34.7 

2  31.9 

6100 

2  4.7 

2  1.9 

110 

4.3 

18.6 

610 

7.2 

15.7 

1200 

0  31.4 

2  33.9 
2  34.9 

6200 

2  8.0 

2  1.7 

120 

4.1 

18.9 

620 

7.3 

15.7 

1300 

0  28.2 

6300 

2  11.2 

2  0.7 

130 

4.0 

19.0 

630 

7.4 

15.6 

1400 

0  25.3 

2  35.0 

6400 

2  14.1 

1  58.8 

140 

4.0 

18.9 

640 

7.5 

15.4 

1500 

0  22.6 

2  34.1 

6500 

2  16.8 

1  56.1 

150 

4.0 

18.8 

650 

7.8 

15.1 

1600 

0  20.1 

2  32  2 

6600 

2  19.3 

1  52.5 

160 

4.2 

18.6 

660 

8.1 

14.7 

1700 

0  17.9 

2  29.5 

6700 

2  21.6 

1  48.3 

170 

4.4 

18.2 

670 

8.4 

14.2 

1800 

0  15.9 

2  25.9 

6800 

2  23.7 

1  43.4 

180 

4.6 

17.7 

6S0 

8.7 

13.5 

1900 

0  14.2 

2  21.5 

6900 

2  25.4 

1  37.8 

190 

4.9 

17.1 

690 

9.2 

12.8 

2000 

0  12.7 

2  16.4 

7000 

2  27.0 

1  31.7 

200 

5.3 

16.5 

700 

9.7 

12.1 

2100 

0  11.5 

2  10.7 

7100 

2  28.2 

1  25.1 

210 

5.7 

15.7 

710 

10.2 

11.3 

2200 

0  10.5 

2  4.4 

7200 

2  29.2 

1  18.2 

220 

6.2 

14.9 

720 

10.7 

10.4 

2300 

0  9.9 

1  57.7 

7300 

2  30.0 

1  11.1 

230 

6.7 

14.1: 

730 

11.2 

9.5 

2400 

0  9.5 

1  50.7 

7400 

2  30.4 

1  3.8 

240 

7.2 

13.2 

740 

11.7 

8.6 

2500 

0  9.4 

1  43.5 

7500 

2  30.6 

0  56.5 

250 

7.7 

12.3 

750 

12.3 

7.7 

2600 

0  9.6 

1  36.2 

7600 

2  30.5 

0  49.3 

260 

8.3 

11.4 

760 

12.8 

6.8 

2700 

0  10.1 

1  28.9 

7700 

2  30.1 

0  42.3 

270 

8.8 

10.5 

770 

13.3 

5.9 

2800 

0  10.8 

1  21.8 

7800 

2  29.5 

0  35.6 

280 

9.3 

9.6 

780 

13.8 

5.1 

2900 

0  11.8 

1  14.9 

7900 

2  28.5 

0  29.3 

290 

9.8 

8.7 

790 

14.3 

4.3 

3000 

0  13.0 

1  8.3 

8000 

2  27.3 

0  23.6 

300 

10.3 

7.9 

800 

14.7 

3.5 

3100 

0  14.6 

1  2.2 

8100 

2  25.8 

0  18.5 

310 

10.8 

7.2 

810 

1.5.1 

2.9 

3200 

0  16.3 

0  56.6 

8200 

2  24.1 

0  14.1 

320 

11.3 

6.5 

820 

15.4 

2.3 

3300 

0  18.4 

0  51.7 

8300 

2  22.1 

0  10.5 

330 

11.6 

5.8 

830 

15.6 

1.8 

3400 

0  20.7 

0  47.5 

8400 

2  19.9 

0  7.8 

340 

11.9 

5.3 

840 

15.8 

1.4 

3500 

0  23.2 

0  43.9 

8500 

2  17.4 

0  5.9 

350 

12.2 

4.9 

850 

16.0 

1.2 

3600 

0  25.9 

0  41.2 

8600 

2  14.7 

0  5.0 

360 

12.5 

4.6 

860 

16.0 

1.1 

3700 

0  28.8 

0  39.3 

8700 

2  11.8 

0  5.1 

370 

12.6 

4.4 

870 

10.0 

1.0 

3800 

0  32.0 

0  38.3 

8300 

2  8.6 

0  6.1 

380 

12.7 

4.3 

880 

15.9 

1.1 

3900 

0  35.3 

0  38.1 

8900 

2  5.3 

0  8.1 

390 

12.8 

4.3 

890 

15.7 

1.4 

4000 

0  38.8 

0  38.7 

9000 

2  1.8 

0  11.1 

400 

12.7 

4.4 

900 

15.4 

1.7 

4100 

0  42.4 

0  40.2 

9100 

1  58.1 

0  15.0 

410 

12.6 

4.6 

910 

15.1 

2.1 

4200 

0  46.2 

0  42.4 

9200 

1  54.3 

0  19.8 

420 

12.5 

5.0 

920 

14.7 

2.7 

4300 

0  50.2 

0  45.4 

9300 

1  50.3 

0  25.3 

430 

12.3 

5.4 

930 

14.3 

3.4 

4400 

0  54.2 

0  49.0 

9400 

1  46.2 

0  31.7 

440 

12.1 

5.8 

940 

13.8 

4.2 

4500 

0  58.4 

0  53.2 

9500 

1  42.0 

0  38.7 

450 

11.8 

6.4 

950 

13.2 

5.0 

4600 

1  2.6 

0  58.0 

9600 

1  37.7 

0  46.3 

460 

11.5 

7.1 

960 

126 

5.9 

4700 

1  6.9 

1  3.1 

9700 

1  33.3 

0  54.3 

470 

11.1 

7.7 

970 

12.0 

6.9 

4800 

1  11.2 

1  8.6 

9S00 

1  28.9 

1  2.7 

480 

10.8 

8.5 

980 

11.4 

7.9 

4900 

1  15.6 

1  14.2 

9900 

1  24.5 

1  11.3 

490 

10.4 

9.2 

9L0 

10.7 

8.9 

5000 

1  20.0 

1  20.0 

1 10000  11  20.0 

i  20.0 

500 

10.0 

10.0  -10(  0 

lO.O 

10.0 

68 


TABLE  XLV. 
Equations  12  to  19 


TABLE  XLVI. 
Equation  20. 


Arg. 

12 

13 

14   15 

16 

17 

18 

19  JArg. 

Arg. 

20 

Arg. 

250 

2.3 

1.6 

7.8 

0.0 

33.7 

3.4 

16.7 

0.4  250 

0 

10.0 

500 

260 

2.3 

1.6 

7.8 

0.0 

33.7 

3.4 

16.7 

0.4  1  240 

10 

10.9 

510 

270 

2.4 

1.7 

7.9 

0.1 

33.6 

3.5 

16.6 

0.4 

230 

20 

11.8 

520 

280 

2.6 

1.9 

8.0 

0.2 

33.5 

3.5 

16.6 

0.5 

220 

30 

12.7 

530 

;290 

2.9 

2.2 

8.2 

0.3 

33.2 

3.6 

16.5 

0.5 

210 

40 

13.5 

540 

300 

3.2 

2.5 

8.4 

0.5 

33.0 

3.7 

16.4 

0.6 

200 

50 

14 -^ 

550 

310 

3.5 

2.9 

8.7 

0.7 

32.7 

3.9 

16.2 

0.7 

190 

60 

15.0 

560 

320 

4.0 

3.4 

9.0 

1.0 

32.4 

4.0 

Ifi.l 

0.8 

180 

70 

15.7 

570 

330 

4.5 

3.9 

9.3 

1.2 

32.0 

4.2 

15.9 

1.0 

170 

80 

16.2 

580 

340 

5.1 

4.4 

9.7 

1.6 

31.6 

4.4 

15.7 

1.1 

160 

90 

16.7 

590 

350 

5.7 

5.1 

10.1 

1.9 

31.1 

4.7 

15.4 

1.3 

1.50 

100 

17.0 

600 

360 

6.4 

5.8 

10.6 

23 

30.6 

4.9 

1.5.2 

1.5 

140 

110 

17.2 

010 

370 

7.1 

6.6 

11.1 

2.7 

30.1 

5.2 

14.9 

1.7 

130 

120 

17.4 

620 

380 

7.9 

7.4 

11.7 

3.2 

29.4 

5.5 

14.6 

1.9 

120 

130 

17.4 

630 

390 

8.7 

8.3 

12.2 

3.6 

28.7 

5.8 

14.3 

2.1 

110 

140 

17.2 

640 

400 

9.6 

9.2 

12.8 

4.1 

28.0 

6.1 

13.9 

2.3 

100 

150 

17.0 

6.50 

410 

10.5 

10.1 

13.5 

4.6 

27.3 

6.5 

13.6 

2.5 

90 

160 

16.7 

660 

420 

11.5 

11.1 

14.1 

5.2 

26,6 

6.8 

13.2 

2.8 

80 

170 

16.2 

670 

430 

12.5 

12.2 

14.8 

5.7 

25.8 

7.2 

12.9 

3.1 

70 

180 

15.7 

680 

440 

13.5 

13.2 

15.5 

6.3 

25.0 

7.6 

12.5 

3.3 

60 

190 

15.0 

690 

450 

14.5 

14.3 

16.2 

6.9 

24.2 

8.0 

12.1 

3.6 

50 

200 

14.3 

700 

460 

15.6 

15.4 

17.0 

7.5 

23.4 

8.4 

11.7 

3.9 

40 

210 

13.5 

710 

470 

16.7 

16.5 

17.7 

8.1 

22.6 

.  8.8 

11.3 

4.1 

30 

220 

12.7 

720 

•480 

17.8 

17.7 

18.5 

8.7 

21.7 

9.2 

10.8 

4.4 

20 

230 

11.8 

730 

490 

18,9, 

18.8 

19.2 

9.4 

20.9 

9.6 

10.4 

4.7 

10 

240 

10.9 

740 

500 

20.0 

20.0 

20.0 

10.0 

20.0 

10.0 

10.0 

5.0 

0 

250 

10.0 

750 

510 

21.1 

21.2 

20.8 

10.6 

19.1 

10.4 

9.6 

5.3 

990 

260 

9.1 

760 

;520 

22.2 

22.3 

21.5 

11.3 

18.3 

10.8 

9.2 

5.6 

980 

270 

8.2 

770 

|530 

23.3 

23.5 

22.3 

11.9 

17.4 

11.2 

8.7 

5.9 

970 

280 

7.3 

780 

540 

24.4 

24.6 

23.0 

12.5 

16.6 

11.6 

8.3 

6.1 

960 

290 

6.5 

790 

,550 

25.5 

25.7 

23.8 

13.1 

15.8 

12.0 

7.9 

6.4 

950 

300 

5.7 

800 

560 

26.5 

26.8 

24.5 

13.7 

15.0 

12.4 

7.5 

6.7 

910 

310 

5.0 

810 

570 

27.5 

27.8 

25.2 

14.3 

14.2 

12.8 

7.1 

6.9 

930 

320 

4.3 

820 

580 

28.5 

28.9 

25.9 

14.8 

13.4 

13.2 

6.8 

7.2 

920 

330 

3.8 

830 

590 

29.5 

29.9 

26.5 

15.4 

12.7 

13.5 

6.4 

7.5 

910 

340 

3.3 

840 

600 

30.4 

30.8 

27.2 

15.9 

12.0 

13.9 

6.1 

7.7 

900 

.350 

3.0 

850 

;610 

31.3 

31.7 

27.8 

16.4 

11.3 

14.2 

5.7 

7.9 

890 

360 

2.8 

860 

•620 

32.1 

32.6 

28.3 

16.8 

10.6 

14.5 

5.4 

8.1 

880 

370 

2.6 

870 

;630 

32.9 

33.4 

28.9 

17.3 

9,9 

14.8 

5.1 

8.3 

870 

380 

2.6 

880 

'640 

33.6 

34.2 

29.4 

17.7 

9.4 

15.1 

4.8 

8.5 

860 

390 

2.8 

890 

i650 

34.3 

34.9 

29.9 

18.1 

8.9 

15.3 

4.6 

8.7 

850 

400 

3.0 

900 

5660 

34.9 

35.6 

30.3 

18.4 

8.4 

15.6 

4.3 

8.9 

840 

410 

3.3 

910 

'670 

35.5 

36.1 

30.7 

18.8 

8.0 

15.8 

4.1 

9.0 

830 

420 

3.8 

920 

'680 

36.0 

36.6 

31.0 

19.0 

7.6 

16.0 

3.9 

9.2 

820 

430 

4.3 

930 

i690 

36.5 

37.1 

31.3 

19.3 

7.3 

16.1 

3.8 

9.3 

810 

440 

5.0 

940 

'700 

36.8 

37.5 

31.6 

19.5 

7.0 

16.3 

3.6 

9.4 

800 

450 

5.7 

950 

710 

37.1 

37.8 

31.8 

19.7 

6,8 

16.4 

3.5 

9.5 

790 

460 

6.5 

960 

720 

37.4 

38.1 

32.0 

19.8 

6,5 

16.5 

3.4 

9.5 

780 

470 

7.3 

970 

730 

37.6 

38.3 

32.1 

19.9 

6,4 

16.5 

3.4 

9.6 

770 

480 

8.2 

980 

740 

37.7 

38.4 

32.2 

20,0 

6,3 

16.6 

3.3 

9.6 

700 

490 

9.1 

990 

:7^0 

37.7 

38.4 

32.2 

20.0 

6.3 

16.6 

3.3 

9.6 

750 

500 

10.0 

1000 

TABLE  XLVII. 


TABLE  XLVIII.  69 


Equations  21  to  29. 


Equations  30  and  31. 


> 

21 

22 

2.^ 

24 

25 

2G 

27 

28 

29 

> 

25 

7.8 

3.2 

7.1 

6.1 

5.9 

4.1 

5.8 

4.3 

5.7 

25 

27 

7.8 

3.2 

7.1 

6.1 

5.9 

4.1 

5.8 

4.3 

5.7 

23 

29 

7.7 

3.3 

7.0 

6.1 

5.9 

t.l 

5.8 

4.3 

5.7 

21 

31 

7.6 

3.3 

7.0 

6.0 

5.8 

4.2 

5.7 

4.3 

5.7 

19 

33 

7.5 

3.4 

6.8 

6.0 

5.8 

4.2 

5.7 

4.4 

5.6 

17 

35 

7.3 

35 

6.7 

5.9 

5,7 

4.3 

5.6 

4.4 

5.6 

15 

37 

7.0 

3.7 

6.5 

5.8 

5.7 

43 

5.6 

4.5 

5.5 

13 

39 

6.8 

3.9 

6.3 

5.7 

5.6 

4.4 

5.5 

4.6 

5.4 

11 

41 

6.5 

4.0 

6.1 

5.6 

5.5 

4.5 

5.4 

4.6 

5.4 

09 

43 

6.2 

4.2 

5.9 

5.5 

5.4 

4.6 

5.3 

4.7 

5.3 

07 

45 

5.9 

4.4 

5.6 

5.3 

5.3 

4.7 

5.2 

4.8 

5.2 

05 

47 

5.5 

4.7 

5.4 

5.2 

5.2 

4.8 

5.1 

4.9 

5.1 

03 

49 

5.2 

4.9 

5.1 

5.1 

5.1 

4.9 

5.0 

5.0 

5.0 

01 

51 

4.8 

5.1 

4.9 

4.9 

4.9 

5.1 

5.0 

5.0 

5.0 

99 

53 

4.5 

5.3 

4.6 

4.8 

4.8 

5.2 

4.9 

5.1 

4.9 

97 

55 

4.1 

5.6 

4.4 

4.7 

4.7 

5.3 

4.8 

5.2 

4.8 

95 

57 

3.8 

5.8 

4.1 

4.5 

4.6 

5.4 

4.7 

5.3 

4.7 

93 

59 

3.5 

6.0 

3.9 

4.4 

4.5 

5.5 

4.6 

5.4 

4.6 

91 

61 

3.2 

6.1 

3.7 

4.3 

4.4 

5.6 

4.5 

5.4 

4.6 

89 

63 

3.0 

6.3 

3.5 

4.2 

4.3 

5.7 

4.4 

5.5 

4.5 

87 

65 

2.7 

6.5 

3.3 

4.1 

4.3 

5.7 

4.4 

5.6 

4.4 

85 

67 

2.5 

6.6 

3.2 

4.0 

4.2 

5.8 

4.3 

5.6 

4.4 

83 

69 

2.4 

6.7 

3.0 

4.0 

4.2 

5.8 

4.3 

5.7 

4.3 

81 

71 

2.3 

6.7 

3.0 

3.9 

4.1 

5.9 

4.2 

5.7 

4.3 

79 

73 

2.2 

6.8 

2.9 

3.9 

4.1 

5.9 

4.2 

5.7 

4.3 

77 

75 

2.2 

6.8 

2.9 

3.9 

4.1 

5.9 

4.2 

5.7  '  4.3 

75 

TABLE  XLIX. 
Equation  32.     Argument,  Siipp.  of  Node. 


o 
0 

Ills 

IVs 

Vs 

Vis 

VII^ 

VIIIs 

3.1 

4.0 

6.5 

10.0 

13.5 

16.0 

o 
30 

2 

3.1 

4.2 

6.8 

10.2 

13.7 

16.1 

28 

4 

3.1 

4.3 

7.0 

10.5 

13.8 

16.2 

26 

6 

3.1 

4.4 

7.2 

10.7 

14.0 

16.3 

24 

8 

3.2 

4.6 

7.4 

11.0 

14.2 

16.4 

22 

10 

3.2 

4.7 

7.6 

11.2 

14.4 

16.5 

20 

12 

3.3 

4.9 

7.9 

11.4 

14.6 

16.6 

18 

14 

3.3 

5.0 

8.1 

11.7 

14.8 

16.6 

16 

16 

3.4 

5.2 

8.3 

11.9 

15.0 

16.7 

14 

18 

3.4 

5.4 

8.6 

12.1 

15.1 

16.7 

12 

20 

3.5 

5.6 

8.8 

12.4 

15.3 

16.8 

10 

22 

3.6 

5.8 

9.0 

12.6 

15.4 

16.8 

8 

24 

3.7 

6.0 

9.3 

12.8 

15.6 

16.9 

6 

26 

3.8 

6.2 

9.5 

13.0 

15.7 

16.9 

4 

28 

3.9 

6.3 

9.8 

13.2 

15.8 

16.9 

2 

30 

4.0 

6.5 

10.0 

13.5 

16.0 

16.9 

0 

lis 

Is 

Os 

XIs 

Xs 

IXs 

Arg. 

30 

31 

0 

5.0 

5.0 

2 

5.0 

50 

4 

4.9 

5.1 

6 

4.9 

5.1 

8 

4.8 

5.2 

10 

4.8 

5.2 

12 

4.7 

5.3 

14 

4.6 

5.4 

16 

4.5 

5.5 

18 

4.4 

5.5 

20 

4.2 

5.6 

22 

4.1 

5.7 

24 

4.0 

5.8 

26 

3.9 

5.8 

28 

3.8 

5.9 

30 

3.7 

5.9 

32 

3.7 

5.9 

34 

3.7 

5.9 

36 

3.7 

5.9 

38 

3.8 

5.8 

40 

3.9 

5.7 

42 

4.1 

5.6 

44 

4.3 

5.5 

46 

4.5 

5.3 

48 

4.8 

5.2 

50 

5.0 

5.0 

52 

5.2 

4.8 

54 

5.5 

4.7 

56 

5.7 

4.5 

58 

5.9 

4.4 

60 

6.1 

4.3 

62 

6.2 

4.2 

64 

6.3 

4.1 

66 

6.3 

4.1 

68 

6.3 

4.1! 

70 

6.3 

4.1! 

72 

6.2 

4.1 

74 

6.2 

4.2 

76 

6.0 

4.2 

78 

5.9 

4.3 

80 

5.8 

4.4 

82 

5.7 

4.5 

84 

5.5 

4.6 

86 

5.4 

4.6 

88 

5.3 

4.7 

90 

5.2 

4.8 

92 

5.1 

4.8 

94 

5.1 

4.9 

96 

5.0 

4.9 

98 

5.0 

5.0 

100 

5.0 

5.0 

Constant  55" 


70 


TABLE   L. 
Evection. 


Argument.      Evection,    corrected. 


0» 


lis 


III' 


IVs 


Diff, 


2o  Diff.  2^ 


7 

8 

9 

10 

11 

12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 


85.5 
85. 4 
85. 4 
85.3 
85.1 


30  00.0 

31  25.5 

32  50. i) 

34  16.3 

35  41.6 

37  6.7 
85.1 

38  31.8  g^  9 

39  56.7^4  7 
4121.4  84  4 
42  45.8  p4  3 

44  10.1 

83.9 

45  34.0  83.7 

82.2 


11.8 
81.3 


52  28.9 

53  50.7 

56  32.9  l''% 

57  53.2,^^-^ 
79.8 

59  13.0  yg  3 

0  32.3  787 

1  51.0  781 

3  9,1  77  4 

4  26.5 

76.8 

5  43.3 

6  59.4 

8  14.9 

9  29.6 
10  43.5 


76.1 
75.5 
74.7 
73.9 


10  43.5 

11  56.7 

13  9.0 

14  20.6 

15  31.3 

16  41.1 


Diflf. 


2° 


Diff. 


17  50.1 

18  58.2 

20  5,3 

21  II"' 65  2 

22  16.7, 

|64.3 

23  21.0 


73.2 
72.3 
71.6 
70.7 
69.8 

69.0 

68.1 
67.1 
66.2 


24  24.2 

1 25  26,4 

126  27,6 

27  27,6 

28  26,6' 

29  24,6 

30  21,4 

31  17,0 

32  11.5, 

33  4.8 

33  57.0 

34  47.9 

35  377 

36  26.2 


37  13.4 

37  59.4 

38  44.2 

39  27.6 

40  9.7 


63.2 
62.2 
61.2 
60.0 

159.0 

58.0 
56.8 
55.6 
54.5 

'53.3 

52.2 
50.9 
49.8 
48.5 

'47.2 

46.0 
44.8 
43.4 
42.1 


[40  9.7 
40  50.6 

[41  30,1 
42  8,3 
42  45.1 

J43  20,6 

;43  54,7 

|44  27,4 

44  58,8 

45  28,7 
,45  57,3 

46  24.5 
146  50.2 

47  14.5 
47  37.4 

47  5S.8 

48  18.8 
48  37.4 

48  54,5 

49  10,1 
49  24,4 

49  37,1 

49  48.3 
49.58,1 
.50  6,4 
.50  13,3 

.50  18,7 

50  22,6 
50  25,0 
50  26,0 
50  25,5 


40.9 
39.5 
38.2 
36.8 
35.5 

34.1 

32.7 
31.4 
29.9 
28.6 

27.2 

25.7 
24.3 
22.9 
21.4 

120.0 

18.6 
17.1 
15.6 
14.3 

12,7 

11,2 

9.8 
8.3 
6.9 

5.4 

3.9 
2.4 
1.0 


0.5 


.50  25.5 
.50  23.5 
50  20.1 
50  15.2 
.50  8.8 
50    1.0 

49  51.7 
49  41.0 
49  28.8 
49  15.1 
49    0.2 

48  43.5 
48  25.6 
48  6.3 
47  45.5 
47  23.3 

46  59.8 
46  34.8  J 
46  8.5' 
15  40.7' 
45  11.6 

44  41.2 
44  9.5 
43  36.4 
43  1.9i 
42  26.21 

41  49.2 
41  10.8 
40  31.2 
39  50.4 
39    8.3 


2.0 
3.4 
4.9 
6.4 

7.8 

9.3 

10.7 
12  2 
13.7 
14.9 

16,7 

17.9 
19,3 
20.8 
22,2 

23,5 
25.0 
26.3 

27.8 
29.1 

30.4 

31.7 
'33.1 
34.5 
35.7 

37.0 

38.4 
39.6 
40.8 
42.1 


23 


39  8.' 
38  24.9 
37  40.4 
36  54.6 
36  7.6 
35  19.5 

34  30.2 
33  39.7 
32  48.1 
31  55.4 
31     1.6 

30  6.7 
29  10.7 
28  13.7 
27  15.7 
26  16.6 

25  16.6 
24  15.6 
23  13.6 
22  10.7 
21     6.8 

20  2.1 
18  56.4 
17  49.9 
16  42.6 
15  34.4 

14  25.5 
13  15.7 
12  5.2 
10  54.0 
9  42.0 


Diff 


43.4 
44.5 
45.8 
47.0 
48.1 

49.3 
50.5 
51.6 
.52.7 
53.8 

54.9 

56.0 
57.0 
58.0 
59.1 

60.0 
61.0 
62.0 
62.9 
63.9 

64.7 

65.7 
66.5 
67.3 
68.2 

68.9 

69.8 
70.5 
71.2 
72.0 


Diff. 


9  42.0 
8  29.3 
7  16.0 
6  2.0 
4  47.4 
3  32,2 

2  16,3 
0  59.9 


59  43.0 
.58  25.6 
57    7.6 

.55  49.2 
.54  30.3 
.53  11.0 
51  51.3 
50  31.2 

49  10.7 
47  49.9 
46  28.8 
45  7.5 
43  45.8 
I 

42  23.9 
41  1.8 
39  39.5 

33  17.0 
36  54.41 

35  31.71 

34  8.8 
32  45.9 
31  23. 0| 
30    O.Oj 


72.7 
73.3 
74.0 
74.6 
75.2 

75.9 

78.4 
76.9 
77.4 
78.0 

78.4 

78.9  1 
79.3 
79.7 
80,1 

80,5 

80,8 
81,1 
81,3 
81,7 

1.9 
82.1 

[82.3 
82.5 
82.6 

82.7 

82.9 
82.9 
82.9 
83.0 


TABLE   L. 

Evection. 


71" 


Argument.     Evection,  corrected. 


Vis 


VIIs 


VIIIs 


IX^ 


XIs 


Diff.O^ 


Diff.O^ 


Diff.  0= 


Diff.  0= 


Diff.  0= 


16 

17  6 

18  5 

19|   4 

20    2 

2ll    1 

22  _0 

23  59 

24  57 

25  56 

26  55 

27  53 

28  52 
29,51 

30:50 

lo° 


0.0 
37.0 
14.1 
51.2 
28.3 

5.6 

43.0 
20.5 
58.2 
36.1 
14.2 

52.5 
31.2 
10.1 
49.3 

23.8 

8.7 
49.0 
29.7 
10.8 
52.4 

34.4 

17.0 

~o 

43.7 

27.8 

12.6 
58.0 
44.0 
30.7 
18.0 


83.0 

82.9 
82.9 
82.9 

82.7 

82.6 

82.5 
82.3 
1 
81.9 

81.7 

81.3 
81.1 
80.8 
80.5 

80.1 

79.7 
79.3 

78.9 
78.4 

78.0 

77.4 
76.9 
76.4 
75.9 

75.2 

74.6 
74.0 
73.3 
72.7 


50  18. 
49  6. 
47  54. 
46  44. 
45  34. 
44  25. 

43  17. 
42  10. 
41  3. 
39  57. 
38  53. 

37  49. 
36  46. 
35  44. 
34  43. 
33  43. 

32  44. 
31  46. 
30  49. 
29  53. 

28  58. 

28  4, 
27  11, 
26  20 
25  29, 
24  40, 

23  52 
23  5 
22  19. 
21  35. 
20  51. 

0° 


72.0 
71.2 
70.5 
69.8 
68.9 

08.2 

67.3 
66.5 
65.7 
64.7 

63.9 

62.9 
62.0 
61.0 
60.0 

59.1 

58.0 
57.0 
56.0 
54.9 

53.8 

52.7 
51.6 
50.5 
49.3 

48.1 

47.0 
45.8 
44.5 
43.4 


20  51.7 
20  9.6 
19  28.8 
18  49.2 
18  10.8 
17  33.8 

16  58.1 
16  23.6 
15  50.5 
15  18.8 
14  48.4 

14  19.3 
13  51.5 
13  25.2 
13  0.2 
12  36.7 

12  14.5 
11  53.7 
11  34.4 
11  16.5 
10  59.8 

10  44.9 
10  31.2 
10  19.0 
10  8.3 
9  59.0 

9  51.2 
9  44.8 
9  39.9 
9  36.5 
9  34.5 

0^ 


42.1 

40.8 
39.6 
38.4 
37.0 

35.7 
34.5 
33.1 
31.7 
30.4 

29.1 

27.8 
26.3 
2o.0 
23.5 

22.2 

20.8 
19.3 
17.9 
16.7 

14.9 

13.7 

12.2 

10.7 

9.3 

7.8 

6.4 
4.9 
3.4 
2.0 


9  34. 
9.34. 
9  35 
9  37. 
9  41. 
9  46. 

9  53. 
10  1. 
10  11. 
10  22. 
10  35. 

10  49. 

11  5, 
1 1  22. 

11  41. 

12  1. 

12  22. 

12  45. 

13  9. 

13  35. 

14  2. 

14  31. 

15  1. 

15  32. 

16  5. 

16  39. 

17  14. 

17  51. 

18  29. 

19  9. 
19  50. 

0^ 


0.5 
1.0 
2.4 
3.9 

5.4 

6.9 

8.3 

9.8 

11.2 

12.7 

|14.3 
15.6 
17.1 
IS. 6 
20.0 

21.4 
22.9 
24.3 
25.7 
27.2 

28.6 

29.9 
31.4 
32.7 
34.1 

35.5 

36.8 
38.2 
39.5 
40.9 


'19  50.3! 

20  32.4 

21  15.8 

22  0.6 
1 22  46.6 
j  23  33.8 

1 24  22.3' 
i25  12.1 
26    3.0 

26  55.2 

27  48.5| 

28  43.0 

29  38.6 

30  35.4 

31  33.4 

32  32.4 

33  .32.4 

34  33.6 

35  .35.8 

36  39.0 

37  43.3 

38  48.5 

39  54.7 

41  1.8 

42  9.9 

43  18.9 

44  28.7 

45  39.4 

46  51.0 

48  3.3 

49  16.5 

O^* 


42.1 
43.4 

44.8 
46.0 
47.2 

I48.5 
49.8 
.50.9 
52.2 
53.3 

54.5 

.55.6 
56.8 
.58.0 
59.0 

60.0 

61.2 
62.2 
63.2 
64.3 

65.2 

G6.2 
67.1 
68.1 
69.0 

69.8 

70.7 
71.6 
72  3 
73.2 


Diff. 


49  16.5 

50  30.4 
5145.1 

53  0.6 

54  16.7 

55  33.5 

56  50.91 
58    9.0 


73.9 

74.7 
75.5 
76.1 
76.8 

!77.4 

78.1 
78.7 


59  27V7Lg 

0  47.0' 


2  6.8 

3  27.1 

4  48.0 
6  9.3 
731.1 
8  53.3 

10  15.9 

11  38.9 

13  2.3 

14  26.0 

15  49.9 

17  14.2 

18  38.6 

20  3.3 

21  28.2 

22  53.3 

24  18.4 

25  43.7 

27  9.1 

28  34.5 
30    0.0 

I'' 


79.8 
80.3 
80.9 
81.3 

81.8 
82.2 

82.6 

83.0 
83.4 
83.7 
83.9 

84.3 

84.4 
84.7 
84.9 
85.1 

185.1 
85.3 

85.4 
85.4 
85.5 


72 


TABLE   LI. 


Equation  of  Moon's  Centre. 
Argument.     Anomaly   corrected. 


0  0 
30 

1  0 
30 

2  0 
30 

3  0 
30 

4  0 
30 

5  0 

30 

6  0 
30 

7  0 
30 

8  0 
30 

9  0 
30 

10  0 

30 

11  0 
30 

12  0 
30 

13  0 
30 

14  0 
30 

15  0 


0* 


Diff       , 
forlOi^O 


0    0  0 

332.6 

7    .5.2 

10  37.8 

1410.3 

1742.7 

21  15.0 
24  47.3 
28  19.4 
3151.2 
35  23.0 

38  54.5 
42  25.8 
45.56.9 
49  27.7 
52  58.2 

.56  28.5 

59  .58.4 

3  2S.0 

6  57.2 

1026.0 

13  54.5 

1722.5 
20  50.1 
2417.3 
3744.0 

31  10.2 
34  35.8 
38  1.0 
41  25.6 
44  49.6 


70  9 
70  9 
70.9 
70.8 
70.8 

70.8 

70.8 
70.7 
70.6 
70.6 

70.5 

70.4 
70.4 
70.3 
70.2 
70.1 

70.0 
69.9 
69.7 
69.6 
69.5 

69.3 
69.2 
69.1 
68.9 

68.7 

68.5 
68.4 
68.2 
68.0 


20  57, 
23 .55 
26  .52 
29  47 
32  42 
35  35 

38  27 
41  18. 
44  7 
46  56 
49  43, 

52  29, 
,55  13 
57  57, 


0  39, 
3  20. 

5  59. 

8  37. 
11  14. 
13.50. 
16  24. 

18  57. 
2123, 
23  .58. 
26  27. 
28  54. 

3120, 
33  44. 
36  7. 
38  29 
40  49. 

11° 


Diff 
forlO 


59.2 
58.9 
58.5 
.58.) 
.57.7 

57.3 

57.0 
.56.5 
56.1 
55.7 

55.3 

.54.9 
54.5 
.54.0 
.53.6 

53.2 

.52.7 
52.3 

51.8 
51.4 

50.9 

.50.5 
.50.0 
49.6 
49.1 

48.6 

48.1 
47.7 
47.2 
46.6 


lis 


12° 


38  43. 
40  14. 
4142. 

43  9. 

44  34. 

45  58. 

47  20. 

48  40. 

49  58. 

51  15. 

52  30. 

53  43. 

54.54. 

56  4. 

57  12. 

58  IS. 

59  22. 

0  2.5; 

126. 

2  25. 

3  23. 

418. 

5  12. 

6  4. 
6.54. 

7  43. 

8:^0. 

9  15. 

9  58. 
1040. 
11  19, 

13° 


Ills 


Diff 
forlOi 


13° 


30.1 
29.6 
29.0 

28.4 
27.8 

j27.3 

ol26.7 
■^  1 26.1 
'12.5.5 
3,2.5.0 

124.4 

23.8 
23.2 
22.6 
22.1 

21.5 

20.9 
20.3 
19.7 
19.1 

18.6 

17.9 
17.4 
16.8 
16.2 

15.6 

I  15  0 
*:14.4 
^  13.8 
^13.3 


IVs 


Diff  ,„. 
for  10  12 


1735.2 
17  20.9 
17  4.8 
1647.1 
16  27.6 
16    6.5 

15  43.7 
15  19.2 
14  53.1 
14  25.2 
13  55.8 

13  24.7 
1251.9 
12  17.4 
1141.4 
11    3.7 

10  24.3 
9  43.4 
9  0.8 
8  16.6 
7  30.8 

6  43.4 
5  54.41 
5  3.9 
411.7 
3  18.0 

2  22.7 

125.8 

0  27.4 

.5927:4 

58  25.9 

12° 


4.8 
5.4 
5.9 
6.5 
7.0 

7.6 

8.2 
8.7 
9.3 
9.8 

10.4 

10.9 
11.5 
12.0 
12.6 

13.1 

13.6 
14.2 
14.7 
15.3 

15.8 

16.3 
16.8 
17.4 
17.9 

18.4 

19,0 
19.5 
20.0 
20.5 


16  20.8 
14  35.3 
12  48.5 
11  0.4 
911.1 
7  20.5 

5  28.7 
3  35.6 
141.3 


59  45.8 
57  49.1 

5551.1 
53  52.0 
5151.7 
49  50.3 
47  47.6 

45  43.8 
43  38.9 
J41  32.8 
139  25.6 
37  17.3 

35    7.9 

32  57.4 
30  45.8 
28  33. 1 
26  19.4 

24   4.6 

2148.8 
1931.9 
1714.1 
14  55.2 

11° 


Ye 


Diff  I, 
forlOi 


Diff 
for  10 


35.2 
35.6 
36.0 
36.4 
36.9 

37.3 

37.7 
38.1 
38.5 
38.9 
39.3 

39.7 
40.1 
40.5 
40.9 

41.3 

41.7 
42.0 
42.4 

42.8 
43.1 

43.5 
43.9 
44.2 
44.6 

44.9 

4.5.3 
45.6 
45.9 
46.3 


|58  28. 
j55  43. 

i52  58. 
.50  11. 
{47  24. 
44  36 

4148 
.38  59 
36  10 
33  19 
30  29 

27  37 
24  45 
2153 
19  0 
16    7 

13  13 

10  18 
7  23 
4  28 
1  32 


^5.5.0 
g55.3 
"55.5 
^55.7 
^|55.9 

56.1 


5«35. 
55  38. 
5241. 
49  43. 
46  45. 

43  47. 
40  48. 

I3749, 
I3449 
13149 


56.3 
56.5 
56.7 
56.9 

57.1 

57.3 
57.5 
57.6 
57.8 

58.0 

58.2 
58.3 
58.5 
58.6 

58.8 

59.0 
59.1 
59.3 
59.4 

59.5 

59.  G 
59.8 
59.9 
60.0 


TABLE  LI. 

Equation  of  Moon's  Centre. 
Argument.      Anomaly  corrected. 


73 


0  0 
30 

1  0 
30 

2  0 
30 

3  0 
30 

4  0 
30 

5  0 

30 

6  0 
30 

7  0 
30 

8  0 
30 

9  0 
30 

10  0 


Yls 


T 


Diff 
for  10 


0    0 


56  54. 
53  49. 
50  43. 
47  38. 
44  33. 

4128. 
38  23. 
35  18. 
32  13. 
29    8. 

26  3. 
22  58. 
19  54. 
16  50. 
13  45. 

1041. 
7  38. 
4  34 
131, 

58  27. 


11  0 
30 

12  0 
30 


30  55  24.9 
2 

7 
5 
6 


13  0 
30 

14  0 
30 

15  0 


52  22. 
4919. 
46  17.: 
43  15. 


4014. 
37  12. 
3411. 
31  10. 

28  10. 

5° 


VIIs 


61.8 
61.8 
61.8 
61.8 
61.7 

61.8 

61,7 
61.7 
61.7 
61.6 
61.6 

61.5 
61.5 
61.4 
61.4 

[61.3 

61.3 
61.2 
61.1 
61.1 

61.0 

60.9 
60.8 
60.7 
60.6 

60.5 

60.5 
60.3 
60.2 
60.1 


4° 


Diff 
forlO 


131.1 
58  46.7 
56  3.0 
53  20.0 
50  37.7 
47  56.2 

4515.4 
42  35.3 
39  56.0 
3717.4 
34  39.6 

32  2.7 
29  26.5 
2651.1 
2416.6 
2142.9 

19  10.0 
16  38.0 
14  6.9 
1136.6 
9    7.3 

6  38.9 
411.3 
144.7 


59  18.9 
56  54.2 

54  30.4 
52  7.5 
49  45.6 
47  24.7 
45   4.8 


54.8 
54.6 
54.3 
.54.1 
.53.8 

53.6 

53.4 
53.1 
.52.9 
.52.6 

.52.3 

52.1 
51.8 
51.5 
51.2 

51.0 

50.7 
50.4 
50.1 
49.8 
49.5 

49.2 
48.9 
48.6 
48.2 

47.9 

47.6 
47.3 
47.0 
46.6 


VIIIs 


43  39.2 
41  55.0 
4012.0 
38  30.5 
36  50.3 
3511.3 

33  33.7 
3157.5 
30  22.6 
28  49.0 
27  16.8 

25  46.1 
2416.7 
22  48.7 
2122.1 
19  56.9 

18  33.1 
1710.8 
1549.8 
14  30.4 
1312.5 

1155.9 

10  40.9 

9  27.3 

8  15.2 

7    4.6 

5  55.4 
4  47.8 
341.7 
2  37.1 
134.1 

1° 


Diff 

for  10 


0° 


34.7 
34.3 
33.8 
:53.4 
33.0 
32.5 

32.1 
31.6 
31.2 
30.7 

30.2 

29.8 
29.3 

28.9 

28.4 

27.9 

27.4 
27.0 
26  5 
26.0 
25.5 

25.0 
24.5 
24.0 
23.5 

23.1 

22.5 
22.0 
21.5 
21.0 


IXs 


42  24.8 
42  12.1 
42  1.2 
41 .52.0 
4144.4 
4138.7 

41  34.6 
41  32.2 
4131.6 
4132.7 
4135.6 

4140.1 
41  46.4 

41  54.5 

42  4.3 
42  15.9 

42  29.2 

42  44.2 

43  1.1 
43  19.6 

43  39.9 

44  2.0 
44  25.9 

44  51.5 
4518.8 

45  48.0 

46  18.9 

46  51.5 

47  26.0 

48  2.2 
48  40.1 

0° 


Xs 


Diff 
for  10 


1° 


4.2 
3.6 
3.1 
2.5 
1.9 

1.4 

0.8 
0.2 
0.4 
1.0 

1.5 

2.1 
2.7 
3.3 
3.9 

4.4 

5.0 
5.6 
6.2 
6.8 

7.4 

8.0 
8.5 
9.1 
9.7 

10.3 

10.9 
11.5 
12.1 
12.6 


21  16.4 

22  48.5 

24  22.2 

25  57.7 
27  34.8 

29  13.7 

30  54.2 
32  36.3 
34  20.2 

36  5.6 

37  52.8 

3941.5 
4132.0 
43  24.0 
45  17.7 
47  12.9 

49  9.8 
51  8.3 
53  8.4 
.55  10.1 
57  13.3 

59  18^ 
T24T5 
3  32.4 
541.9 
752.9 

10  5.5 
12  19.5 
1435.1 
1652.1 
1910.7 

2° 


XI" 


Diff" 
for  10 


30.7 
31.2 
31.8 
32.4 
33.0 

33.5 

34.0 
34.6 
35.1 
35.7 

36.2 

36.8 
37.3 
37.9 

38.4 

39.0 

39.5 
40.0 
40.6 
41.1 
41.6 

42.1 
42.6 
43.2 
43.7 

44.2 

44.7 
4.5.2 
45.7 
46.2 


3° 


57^10.7 

0  15.8 
321.8 
6  28.8 
9  36.8 

1245.7 
1555.5 
19  6.2 
22  17.8 
25  30.3 

2843.7 
31  57.8 
35  12.9 
38  28.7 
4145.2 

45  2.6 
48  20.7 
51  39,6 
54  59.1 
5819.3 


140.3 
5    1.9 

8  24.1 
1146.9 
15  10.4 

5° 


Diff" 
for  10 


59.6 
60.0 
60.3 
60.7 
61.0 
61.3 

61.7 
62.0 
62.3 
62.7 

63.0 

63.3 
63.6 
63.9 
64.2 

64.5 

64.7 
65.0 
65.3 
65.5 

65.8 

66.0 
66.3 
66.5 
66.7 

67.0 

67.2 
67.4 
67.6 
67.8 


74 


TABLE    LI. 
Equation  of  Moori's  Centre, 
Argument.     Anomaly  corrected. 


Os 


Us 


Ills 


IVs 


Ys 


Diff 
folio 


11- 


Difr 

forlO 


13^ 


Diff 

forlO 


15  0,44  49.6 
30  48  13.1 

16  0|5135.9 
30,54. 58.1 

17  0  58  19.7 
140.7 


30 


18  0 
30 

19  0 
30 

20  0 

30 

21  0 
30 

22  0 

30 

I 

23  0 
30 

24  0 
30 

25  0 

30 

26  0 
30 

27  0 
30 

28  0 
30 

29  0 
30 

30  0  20  57.9 

I 

Il0= 


5    0.9 

8  20.4 

1139.3 

14  57.4 

18  14.8 

2131.3 

24  47.1 
28  2.2 
31  16.3 
34  29.7 

37  42.2 
40  53.8 
44    4 
4714.3 
50  23.2 

5331.2 

56  39.2 
59  44.2 


2  49.3 
5  53.3 

8  56.3 
1158.3 
14  593 
17  59.2 


67.8 
67.6 
67.4 
67.2 
67.0 
66.7 

66.5 
66.3 
66.0 
65.8 

65.5 

65.3 
65.0 
64.7 
64.5 

64.2 

63.9 
63.6 
63.2 
63.0 

62.7 

62.3 
62.0 
61.7 
61.3 

61.0 

60.7 
60.3 
60.0 
59.6 


40  49.3 
43  7.9 
45  24.9 
47  40.5 
49  54.5 
52    7.1 

5418.1 
56  27.6 
58  35.5 

olTs 

2  46.7 

4  49.9 
651.6 
851 
10  50.2 
12  47.1 

1442.3 
16  36.0 
18  28.0 
20  18.5 

22  7.2 

23  54.4 
25  39.8 
27  23.7 

29  5.8 

30  46.3 

32  25  2 

34  2  3 

35  37.8 

37  11.5 

38  43.6 

12° 


46.2 
45.7 
45.2 
44.7 
44.2 

43.7 

43.2 
42.6 
42.1 
41.6 

41.1 

40.6 
40.0 
39.5 
39.0 

38.4 

37.9 
37.3 
36.8 
36.2 

35.7 

35.1 
34.6 
34.1 
33.5 
33.0 

32.4 
31.8 
31.2 
,30.7 


11  19.9 
1157.8 

12  34.0 

13  8.5 
1341.1 
1412.0 

1441.2 
15  8.5 
15  34.1 

15  58.0 

16  20.1 

16  40.4 

16  58.9 
1715.8 

17  30.S 
17  44.1 

17  55.7 

18  5.5 
18  13.6 
18  19.9 
18  24.4 

18  27.3 

18  28.4 
18  27.8 
18  25.4 
1821.3 

118  15.6 
118  80 
i  17  58.8 
1747.9 
17  35.2 

13= 


12. G 
12.1 
11.5 
10.9 
10.3 

9.7 

9.1 

8.5 
8.0 
7.4 

6.8 

6.2 

5.6 
5.0 
44 

3.9 

33 

2.7 
2.1 
1.5 

1.0 

0.4 
0.2 
0.8 
1.4 
1.9 

2.5 
3.1 
3.6 
4.2 


12- 


Diff 
for  10 


11^ 


Diff 
forlO 


58  25.9 
.57  22  9 
.56  18.3 
55  12.2 
54  4.6 
52  55.4 

51  44.8 
.50  32.7 
49  19.1 
48  4.1 
46  47.5 

45  29.6 
4410.2 
42  49.2 
4126.9 
40    3.1 

38  37.9 
37  11.3 
35  43  3 
34  13.9 
32  43.2 

31  11.0 
29  37.4 

28  2.5 
26  26.3 

24  48 

23  9.7 
21  29.5 
19  48.0 
18  5.0 
16  20.8 

12° 


21.0 
21.5 
22.0 
22.5 
23.1 

23.5 

24.0- 
24.5 
25.0 
25.5 

26.0  I 

26.5 
27.0 

27.4 
27.9 

28.4 

28.9 
29.3 
29.8 
30.2 

30.7 

31.2 
31.6 
32.1 
32.5 

33.0 

33.4 
33.8 
34.3 
34.7 


14  55  2 

12  35.3 

10  14.4 

7  52.5 

5  29.6 

3    5.8 

O^Ll 
58  15.3 
55  48.7 
.5321.1 
50  52.7 

48  23.4 
45  53.1 
43  22.0 
40  50.0 
38  17.1 

35  43  4 
33  8.9 
30  33.5 
27  57.3 
25  20.4 

122  42.6 
20  4.0 
17  24.7 
14  44.7 
12    3.8 


46.6 
47.0 
47.3 
47.7 
47.9 

48.3 

48.6 
48.9 
49.2 
49.5 

49.8 

.50.1 
50 
50 
51.0 

51.2 

51.5 
51.8 
.52.1 
52.3 

52.6 

.52.9 

53.1 

3.3 

53.6 

53.8 
9  22.3,.  , 
6  40.0,^f  I 
3  57.0 
1  13  3 


Diff 
for  10 


3149.4 
28  49.1 
25  48.4 
22  47.4 
19  46.0 
16  44.4 

13  42.5 

10  40.3 

7  37.8 

4  35.1 

132.2 


1.58  28.9 
\9° 


54.3 
.54.6 

54.8 


.58  29.0 
55  25.6 
.52  22.0 
49  18.1 
46  14.2 

43  10.0 
40  5.7 
37  1.2 
33  56.6 
30  51.9 

27  47.0 
2442.0 
21  37.0 
1831.8 
15  26.6 

1221.4 
9  16.1 
6  10.8 
3  54 
0    0.0 

70 


PO.l 
ro.2 
60.3 
60.5 
60.5 

60.6 

60.7 
60.8 
60.9 
61.0 
61.1 

61.1 
61.2 
61.3 
61.3 

61.4 

61.4 
61.5 
61.5 
61.6 

61.6 

61.7 

n.7 

^il.7 
61.7 

61.7 

61.8 
61.8 
61.8 
61.8 


TABLE  LI. 

Equation  of  Moon's  Centre. 
Argument.      Anomaly  corrected. 


76 


70 


TABLE   LII. 
Variation. 

Arsument.     Variation,  corrected. 


0» 


lis 


Ills 


IVs 


V» 


Diff.  P 


Diff.  1 


Diff.  Oo 


Diff. 


0° 


Diff.  0° 


Diff. 


0  38 
139 
2J40 
3'41 

4 '42 
544 


6 
7 
8 
9 
10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 


26  5 

27  6 

28  6 

29  7 

30  8 

1° 


0.0 
13.3 

26. 5j 
39.5 

52.2 
4.5 

16.4 
27.7 
38.4 
48.3 
57.4 

5.6 
12.8 
18.9 
23.S 
27.5 

29.8 
30.7 
30.1 
28.0 
24.2 

18.7 
11.4 
2.3 
51.2 
38.2 

23.1 

6.0 

46.7 

25.2 

1.5 


73.3 
73.3 
73.0 
72.7 
72.3 
71.9 

71.3 
70.7 
69.9 
69.1 

68.2 

67.2 
66.1 
64.9 
63.7 

62.3 

60.9 
59.4 
57.9 
56.2 
54.5 

52.7 
50.9 
48.9 
47.0 
44.9 

42.9 
40.7 
38.5 
36.3 


8    1.5 

8  35.5 

9  7.2 
9  36.5 

10  3.4 
10  27.9 

10  49.9 

11  9.4 
11  26.4 
11  40.9 

11  52.9 

12  2.2 
12  9.0 
12  13.2 
12  14.8 
12  13.9 

12  10.3 
12  4.2 
11  55.5 
11  44.2 
11  30.5 

11  14.1 
10  55.3 
10  34.0 
10  10.2 
9  44.0 

9  15.4 
8  44.5 
8  11.2 
7  35.7 
6  57.9 

1° 


34.0 

31.7 
29.3 
26.9 
24.5 

22.0 

19.5 
17.0 
14.5 
12.0 

9.3 

6.8 
4.2 
1.6 
0.9 

3.6 

6.1 

8.7 

11.3 

13.7 

16.4 

18.8 
21.3 
23.8 
26.2 

28.6 

30.9 
33.3 
35.5 
37.8 


6  57.9' 
6  18.0 
5  35.9 
4  51.7 
4  5.5 
3  17.3 

2  27.21 
1  35.3 
041.6 


59  46.1 
58  49.0 

57  50.2' 
56  50.0 
55  48.3' 
54  45.2' 
53  40.9 

52  35.3 

51  28.5! 
^50  20.71 
49  11. 9i 

48    2.2 

:46  51.7i 
45  40.5 
44  28.6' 
43  16.1. 

(42    3.2 

'4O49.9I 
39  36.2 
38  22.4 
37  8.4 
35  54.4 

03 


39.9 

42.1 
44.2 
46.2 
48.2 

50.1 

51.9 
53.7 
55.5 
57.1  I 

|58.8 

60.2 
61.7 
63.1 
64.3 

165.6 

66.8 
67.8 
68.8 
69.7 

70.5 

71.2 
71.9 
72.5 
72.9 

73.3 

73.7 
73.8 
74.0 
74.0 


35  54 
34  40 
33  26 
32  13 
30  59 
29  46 

28  34 
27  22 
26  11 
25  0 
23  51. 

i22  42. 

:2i  34. 

:20  27. 
!l9  22. 

|18  18. 

17  15. 
|l6  13. 

15  13. 

14  14. 
il3  17. 

12  22. 
11  28. 
10  36. 
i   9  46. 

i   8  58. 

8  12. 
7  28. 
6  46. 
6  7. 
5  29. 


Oo 


74.0 
73.8 
73.6 
73.4 
72.9 
72.4 

71.9 
71.2 
70.5 
69.6 

68.8 

67.8 
66.6 
65.6 
64.3 
63.0 

61.6 
60.2 
58.6 
57.1 

55.3 

53.7 
51.8 
49.9 
48.0 
46.1 

44.0 
41.9 
39  7 
37.6 


5  29.5 
4  54.2 
421.3 
3  50.6 
3  22.3 
2  56.5 

2  33.1 
2  12.1 
1  53.7 
1  37.8 
1124.5 

|l  13.7 
,1  5.6 
1  0.0 
0  57.0 
0  56.7 

0  59.0 

1  3.9 
1  11.5 
1  21.6 
134.4 

1  49.8 

2  7.8 
2  28.3 

2  51.4 

3  16.9 

3  45.0 

4  15.6 

4  48.5 

5  23.9 

6  1.6 

0^ 


35.3 
32.9 
30.7 
28.3 
25.8 
23.4 

21.0 
18.4 
15.9 
13.3 

10.8 

8.2 
5.5 
3.0 
0.3 

2.3 

4.9 

7.6 

10.1 

12.8 

15.4 

18.0 
20.5 
23.1 
25.5 
28.1 

30.6 
32.9 
35.4 
37.7 


6  1.6 
641.6 

7  23.9 

8  8.4 

8  55.0 

9  43.7 

10  34.5 

11  27.3 

12  22.0 

13  18.6 

14  16.9 

15  17.0 

16  18.7 

17  22.0 

18  26.9 

19  33.1 

20  40.7 

21  49.6 

22  59.6 

24  10.8 

25  22.9 

26  35.9 

27  49.8 

29  4.5 

30  19.7 

31  35.6 

32  51.9 

34  8.6 

35  25.6 

36  42.7 
38    0.0 

0° 


40.0 
42.3 
44.5 
46.6 
48.7 

508 

52.8 
54.7 
56.6 
58.3 

60.1 

61.7 
63.3 
64.9 
66.2 

67.6 

68.9 
70.0 
71.2 
72.1 
73.0 

73.9 
74.7 
75.2 
75.9 

76.3 

76.7 
77.0 
i77.1 
77.3 


TABLE   LII. 

Variation. 

Argument,     Variation  corrected. 


77 


VI* 


VIIs 


VIII* 


IX* 


XI* 


^0^ 


Diff.  1° 


38  0. 

39  17. 

40  34. 
4151. 

43  8. 

44  24. 

45  40. 

46  55. 

48  10. 

49  24. 

50  37. 

51  49. 

53  0. 

54  10. 

55  19. 

56  26. 

57  33. 

58  38. 

59  41. 
0  43. 
143. 

241. 
3  38. 
432 

5  25 

6  16 

7  5 

7  51 

8  36 
918 


30    9  58.4 
1° 


77.3 
77.1 
77.0 
76.7 
76.3 

75.9 

75.2 
74.7 
73.9 
73.0 

72.1 

71.2 
70.0 
68.9 
67.6 

66.2 

64.9 
63.3 
61.7 
60.1 
58.3 

56.6 
54.7 
52.8 
50.8 

48.7 

46.6 
44.5 
42.3 
40.0 


9  58.4 

10  36.1 

11  11.5 

1 1  44.4 

12  15.0 
1243.1 

13  8.6 
1331.7 

13  52.2 
1410.2 

14  25.6 

14  38,4 
1448.5 

14  56.1 

15  1.0 
15    3.3 

15    3.0 

15  0.0 
14  54.5 
14  46.3 
14  35.5 

14  22.2 
14  6.3 
13  47.9 
13  26.9 
13    3.5 

12  37.7 
12  9.4 
11  38.7 
11  5.8 
10  30.5 


Diff. 


1° 


Diff.  Oo  Diff.  0°         Diff 


37.7 
35.4 
32.9 
30.6 
28.1 

25.5 

23.1 
20.5 
18.0 
15.4 
12.8 

10.1 
7.6 
4.9 
2.3 

0.3 

3.0 

5.5 

8.2 

10.8 

13.3 

15.9 
18.4 
21.0 
23.4 

25.8 

28.3 
30.7 
32.9 
35.3 


10  30.5 
9  52.9 
9  13.2 
831.3 
7  47.3 
7    1.2 

6  13.2 
5  23.3 
431.5 
3  37.8 
2  42.5 


7.6 
39.7 
41.9 
44.0 
46.1 

48.0 

49.9 
51.8 
53.7 
55.3 

57.1 

58.6 
60.2 
61.6 
63.0 
64.3 

65.6 
66  6 
67.8 
68.8 

69.6 

70.5 
71.2 
71.9 
72.4 

72.9 

73.4 
73.6 
73.8 
74.0 


0° 


40  5.6 
3851.6 
37  37.6 
30  23.8 
35  10.1 
33  56.8 

32  43.9 
3131.4 
30  19.5 
29  8.3 
27  57.8 

26  48.1 
25  39.3 
2431.5 
23  24.7 
22  19.1 

21  14.8 
20  11.7 
19  10.0 
18  9.8 
1711.0 

16  13.9 
15  18.4 
14  24.7 
13  32.8 
12  42.7 

1154.5 

11    8.3 

1024.1 

9  42.0 

9    2.1 

Oo 


74.0 
74.0 
73.8 
73.7 
73.3 

72.9 

72.5 
71.9 
71.2 
70.5 

69.7 

68.8 
67.8 
66.8 
65.6 

64.3 

63.1 
61.7 
60.2 
58.8 

57.1 

55.5 
53.7 
51.9 
50.1 

48.2 

46.2 
44.2 
42.1 
39.9 


9  2.1 
8  24.3 
7  48.8 
7  15.5 
6  44.6 
6  16.0 

5  49.8 
5  26.0 
5  4.7 
4  45.9 
4  29.5 

4  15.8 
4  4.5 
3  55.8 
3  49.7 
3  46.1 

3  45.2 
3  46.8 
3  51.0 

3  57.8 

4  7.1 

4  19.1 
4  33.6 

4  50.6 

5  10.1 
5  32.1 

5  56.6 

6  23.5 

6  52.8 

7  24.5 
7  58.5 

0° 


0° 


Diff 


37.8 
35.5 
33  3 
30.9 
28.6 

26.2 

23.8 
21.3 

18.8 
16.4 

13.7 

11.3 

8.7 
6.1 
3.6 

0.9 

1.6 
4.2 
6.8 
9.3 

12.0 

14.5 
17.0 
19.5 
22.0 
24.5 

26.9 
29.3 
31.7 
34.0 


7  58.5 

8  34.8 

9  13.3 
9  54.0 

10  36.9 
1121.8 

12    8.8 

12  57.7 

13  48.6 
1441.3 

15  35.8 

16  32.0 
1729.9 

18  29.3 

19  30.2 

20  32.5 

2136.2 
2241.1 

23  47.2 

24  54.4 

26  2.6 

27  11.7 
2821.6 

29  32.3 

30  43.6 
3155.5 

33  7.8 

34  20.5 

35  33.5 

36  46.7 
38    0.0 

0° 


36.3 

38.5 
40.7 
42  9 
44.9 

47.0 
48.9 
50.9 
52.7 
54.5 

56.2 

57.9 
59.4 
60.9 
62.3 
63.7 

64.9 
66.1 
67.2 
68.2 
69.1 

69.9 
70.7 
71.3 
71.9 

72.3 

72.7 
73.0 
73.2 
73.3 


78  TABLE  LIII.     Reduction. 

Argument.     Supplement  of  Node  +  Moon's  Orbit  Longitude. 


0.0 
45.6 
31.2 
16.9 

2.6 
48.4 

34.3 
20.3 
'6.4 
52.6 
39.0 

25.6 
12.3 
59.3 
46.5 
33.9 

21.6 
9.5 
57.7 
46.2 
35.0 

24.2 
13.7 
3.5 
53.7 
44.2 

35.2 

26.5 

183 

10.4 

3.0 


14.4 
14.4 
14.3 
14.3 
14.2 
14.1 

14.0 
13.9 
13.8 
13.6 

13.4 

13.3 
13.0 
12.8 
12.6 

12.3 

12.1 
11.8 
11.5 
11.2 
10.8 

10.5 

10.2 

9.8 

9.5 

9.0 

8.7 
8.2 
7.9 
7.4 


\s  VIIsDiff.    IlsVIIIs   Difir. 


3  0 

7  0 
56.0  '■" 
.„  ,  0.0 
49.0 


43.4 
37.8 
32.7 

28.2 
23.9 
20.0 
16.8 
14.1 

11.8 
10.1 

8.8 
8.1 

7.8 

8.1 

8.8 


0 

lo.i! 

0 

11.8 

0 

14.1 

0 

16.8 

0 

20.0 

0 

23.9 

0 

28.2 

0 

32.7 

0 

37.8 

0 

43.4 

0 

49.5 

0 

56.0 

1 

3.0 

6.1 
5.0 
5.1 

4.5 

4.3 
39 
32 

27 

2.3 

1.7 
1.3 
0.7 
0.3 

l0.3 

0.7 
1.3 
1.7 
2.3 

2.7 

3.2 
3.9 
4.3 
4.5 

5.1 

5.6 
6.1 
6.5 
7.0 


3.0 
10.4 
18.3 
20.5 
35.2 


1  44.2 

1  53.7 

2  35 
2  13.7 
2  24.2 
2  35.0 

2  46.2 

2  57.7 

3  9.5 
3  21.6 
3  33.9 

3  46.5 

3  59.3 

4  12.3 
4  25.6 
4  39.0 


52.6 

6.4 

20.3 

34.3 

48.4 

2.6 
16.9 
31.2 
45.6 

0.0 


7.4 
7.9 
8.2 
8.7 
9.0 
9.5 

9.8 
10.2 
10.5 
10.8 

11.2 

11.5 
11.8 
12.1 
12.3 
12.6 

12.8 
13.0 
13.3 
13.4 
13.6 

13.8 
13.9 
14.0 
14.1 

14.2 

14.3 
14.3 
14.4 
14.4 


7  0.0 
7  14.4 
7  28.8 
7  43.1 

7  57.4 

8  11.6 

8  25.7 
8  39.7 

8  53.6 

9  7.4 
9  21.0 


9  34.4 

9  47.7 

10    0.7 

10  13.5 

10  26.1 


10  38.4 

10  50.5 

11  2.3 
11  13.8 
11  25.0 

11  35.8 
11  46.3 

11  56.5 

12  6.3 
12  15.8 

12  24.8 
12  33.5 
12  41.7 
12  49.6 
12  57.0 


Diff. 


14.4 
14.4 
14.3 
14.3 
14  2 
14.1 

14.0 
13.9 
13.8 
13.6 

13.4 

133 
13.0 
12.8 
12.6 

12.3 

12.1 
11.8 
11.5 
11.2 
10.8 

10.5 

10.2 

9.8 

9.5 

9.0 

8.7 
8.2 
7.9 
7.4 


IVs  Xs    Diff.    \s  XIs      Diff. 


2  57.0; 

3  4.0 
3   105 

16.6 
22.2 
27.3 

31.81 
36  II 
40.0 
43.2 
45.9 


48.2 
49.9 
51.2 
51.9 
52.2 

51.9 
51.2 
49.9 
48.3 
45.9 

43.2 
40.0 
36.1 
31.8 
27.3 

22.2 
16.6 
10.5 
4.0 
57.0 


7.0 
6.5 
6.1 
5.6 
5.1 

|4.5 

4.3 
3.9 
3.2 

2.7 

2.3 

1.7 
1.3 
0.7 
0.3 

0.3 

0.7 
1.3 
1.7 
2.3 

2.7 

32 

39 
4.3 
4.5 

5.1 

5.6 
6.1 
6.5 
7.0 


12  57.0 

12  49.6 

12  41.7; 

12  33.5 

12  24.8 

12  15.8, 


6.3 

56.5 
46.3 
35.8 
25.0 

13.8' 
2.3 
10  50.5 
10  38.4 
10  26.1 


135 
0.7 
47.7 
34.4 
21.0 

7.4 
53.6 
39.7 
25.7 
11.6 

57.4 
43.1 
28.8 
14.4 
0.0 


7.4 
7.9 
8.2 
8.7 
9.0 
9.5 

9.8 
10.2 
10.5 
10.8 
,11.2 

11.5 
11.8 
12.1 
12.3 

|12.6 

12.8 
13.0 
13.3 
13.4 
13.6 

13.8 
13.9 
14.0 
14.1 
14.2 

14.3 
14.3 
14.4 
14.4 


TABLE  LIV.     Lunar  Nutation  in  Longitude. 
Argument.     Supplement  of  the  Node. 


Os 

h 

lis 

Ills 

IV* 

Ys 

+ 

+ 

+ 

+ 

+ 

+ 

° 

0 

0.0 

8.5 

14.8 

17.3 

15.2 

8.8 

o 
30 

2 

0.6 

9.0 

15.1 

17.2 

14.9 

8.1 

28 

4 

1.2 

9.4 

15.4 

17.2 

145 

7.7 

26 

6 

1.7 

10.0 

15.6 

17.2 

14.2 

7.2 

24 

8 

2.3 

10.4 

15.9 

17.2 

13.8 

6.5 

22 

10 

2.9 

10.9 

16.4 

17.1 

13.5 

6.1 

20 

12 

3.5 

11.4 

16.3 

17.0 

13.0 

5.4 

18 

14 

4.1 

11.8 

16.5 

16.9 

12.6 

4.8 

16 

16' 

4.6 

12.2 

16.7 

16.7 

12.2 

43 

14 

18 

5.2 

12.6 

16.8 

16.5 

11.8 

3.7 

12 

20 

5.8 

13.1 

16.9 

16.4 

11.3 

3.0 

10 

22 

6.2 

13.4 

17.1 

16.2 

10.9 

2.4 

8 

24 

6.9 

13.8 

17.1 

15.9 

10.4 

1.8 

6 

26 

7.4 

14.1 

17.2 

15.7 

9.8 

1.3 

4 

28 

7.8 

14.5 

17.2 

15.4 

9.4 

0.6 

2 

30 

8.5 

14.8 

17.3 

15.2 

8.8 

0.0 

0 

xr« 

Xs 

IX* 

VIII« 

VTI* 

VI. 

TABLE  LV. 


79 


Moon's  Distance  from  the  North  Pole  of  the  Ecliptic. 
Argument.     Supplement  of  Node  +  Moon's  Orbit  Longitude. 


r 

Ills 

\Ys 

Ys 

Vis 

YUs 

VIII* 

84° 

85° 

Diff 

for  10 

87° 

Diff.  '„o 
for  10  **^ 

1 

Diff. 

for  10 

000       !  Diff 

^■*          jfor  10 

94° 

O        ' 

0  0 

39  16.0 

'    "    \  „ 
20  42.7  „^  „ 

1346.6 

,"48   0.0 

" 

22  13.4' /'  r. 

265224"-^ 
29  10.2l^-g 

15  17.3 

0    ' 
30  0 

30 

39  16.7  22    4.2.;'-^ 
39  18.8  -23  27.0.;'" 
39  22  4  24  51.0  iZi 

16    6.9 

.«  «   .50  41.4  ,„  0 

16  37.7 

30 

1  0 

18  27.8  ?:";53 22.9  ^oo 

17  56.8 

29  0 

30 

20  49.5^'-^  .56    4.3 
23  11.8*^*!  58  45.7 

.53.8 
53.8 

19  14.6 

30 

2  0 

39  27.3  26  16.2 

28.8 

20  31.3 

28  0 

30 

39  33.7  27  42.6 

29.2 

25  34.8 

47.9 

127.0 

53.8 

33  44.2 

145.3 

21  46.7 

30 

3  0 

3941.5  29  10.1 

29.  i; 

27.58.5 

48.1 

AO    0 

4    8.3 

53.7 
53.7 
.53.7 
53.6 

36    0.2V  n 
38  15.3j-« 
40  29.7^° 
42  43.3.^3 
44  56.2r^-^ 

23    0.8 

27  0 

30 

39  50.6  30  38.9 

30  22.8 

6  49.5 

2413.7 

SO 

4  0 

40    1.2j32    8.8 

•      •^2  47  7'*°"' 

9  30.6 

25  25.3 

26  0 

30 

40  13.2  33  39.9 

30.8 

35  13  2;^^  " 
37  39.3*^-^ 

1211.6 

26  35.7 

30 

5  0 

40  26.7 

35  12.2 

14  52.5 

27  44.8 

25  0 

31.1 

48.9 

53.6 

44.0 

30 

40  41.5 

36  45  6 

31.5 

40    6.1 

49.1 

An  0 

17  33  3 

.53.6 
.=.3.5 
53.5 
.03.4 

47    8.1  .„„ 
49  19.4ig-^ 
5129.71,.,^ 

28  52.6 

30 

6  0 

40  57.7 

38  20.1 

42  33.4 

20  14.0 

29  59.0 

24  0 

30 

41  15.4 

39.55.81:^;-:; 
4132.7|oo'^ 

45    1.2:^^ 

22  .54  4 

31    4.3 

30 

7  0 

4134.4 

47  29.6 

49.7 

25  34.8 

53  39.3 

42.9 

32    8.2 

23  0 

30 

41  54.8 

43  10.6 

o^.v 

49  58.6 

28  14.9 

55  48.0 

33  10.9 

30 

33.0 

49.8 

.53.3 

42  6 

8  0 

4216.7 

44  49.7 

334 
33.8 
34.1 
34.5 

.52  28.1 

50.0 
.50.2 
50  4 

30  54.9 

,53.3 
53.2 
.53.1 
53.0 

.57 .55.8 

42  3 
42.0 
41.7 
41.5 

3412.2 

22  0 

30 
.9  0 

42  39.9  46  29.9 

43  4.6  48  11.2 

54.58.2 
57  28.7 

33  34.7 
36  14.3 

0    2.8 
2    8.9 

35  12.2 

36  10.9 

30 
21   0 

30 

43  30.6  49  53.5 

.59  59.8 

38  53.7 

414.1 

37    8.3 

30 

10  0  '43  58.115137.0 

~2'31.3 

^"•^  '41  32.8 

6  18.4 

38    4.4 

20  0 

34.9 

50.7 

53.0 

■ 

41.1 

30 

44  26.9 

.53  21.6 

35.2 
35.7 
35.9 
36.2 

5    3.3 

?;?t9  28:7 
,5444.6 

52.9 

.52.8 
.52.7 
.52.6 

8  21.8 

40.8 
40.5 
40.2 

on  n 

38  59.1 

30 

11  0 
30 

12  0 

44  57.1 

45  28.8 

46  1.8 

.55    7.1 
.56  53.8 
.58  41.6 

7  35.8 
10    8.8 
1242.1 

10  24.3 
12  25.9 
1426.6 

39  52.5 
4044.6 
41  35.3 

19  0 
30 

18  0 

30 

46  36.1 

1)3073 

15  16.0 

16  26.3 ''''•^ 

42  24.7 

30 

36.6 

51.4' 

52.5 

139.6 

13  0 

4711.9 

2  20.1 

37.0 
37.3 
37.6 
38.0 

1750.2 

^15722.1 
•"•l-S  59.59.3 
m     236.2 

5    9     ^12-^ 
^^■^     748.9 

.52.4 
.52.3 
.52.2 
52.1 

18  25.0' 

20  22.8^^3 

22  19.7  ^«-^ 
2415.5  38  6 

4312.7 

17  0 

30 
14  0 

47  49.0'   411.0 

48  27.5    6    2.9 

20  24.9 
22  .59.9 

43  59.4 

44  44.7 

30 
16  0 

30 

49    7.4    7  55.7 

25  35.3 

45  28.7 

30 

15  0 

49  48.7    9  49.6 

28  11.1 

26  10.4 

^KD.O 

4611.3 

15  0 

84=        86= 

88° 

91° 

93° 

94° 

II* 

Is 

0" 

XI» 

X« 

IX« 

80  TABLE  LV. 

MoorCs  Distance  from  the  Noi'th  Pole  of  the  Ecliptic- 
Argument.     Supplement  of  Node+Moon's  Orbit  Longitude. 


Ills 


84° 


15  049  48.7 
30  50  31.3 

16  0  51  15.3 
30  52    0.6 

17  0  52  47.3 
30  53  35  3 

18  0  54  24.7 
30  55  15  4 

19  0  56    7.5 
30  57    0.9 

20  0  57  55.6 

30  58  51.7 

21  0  59  491 
30    0  47  8 

22  0    1  47.8 
30    2  49.1 


IVs 


23  0 
30 

24  0 
30 

25  0 


351.8 
4  55.7 

6  1.0 

7  7.4 

8  15.2 


30    9  24.3 

26  OJIO  34.7 
30,11  46.3 

27  012  59.2 
30  14  13.3 

28  0  15  28.7 
30  16  45.4 

29  0  18    3.2 
30  19  22.3 

30  0  20  42.7 


86° 

9  49  6 
11  44  5 
13  40  3 
15  37.2 
17  35.0 
19  33.7 

21  33.4 
23  34  1 
25  35.7 
27  38.2 
29  41.6 

31  45.9 
33  51.1 
35  57.2 
38  4.2 
40  12.0 

42  20  7 
44  30  3 
46  40.6 
48  51.9 
51    3.8 

53  16.7 
55  30  3 

1 57  44.7 
59  59.8 


2  15.8 

4  32  5 

6  49.8 

9    7.8 

11  26  9 

13  46.6 


Difi: 

for  10 


85° 


87° 


II« 


38.3 
:J8.6 
39.0 
39.3 
39.6 

39.9 

40.2 
40.5 
40.8 
41.1 

41.4 

41.7 
42.0 
42.3 
42.6 

42.9 

43.2 
43.4 
43.6 
44.0 

44.3 

44.5 
44.8 
45  0 
45  3 
45.6 

45.8 
46.0 
46.4 
46.6 


28  11.1 

30  47.3 

33  23  8 

36  0.7 

38  37.9 

41  15.4 

43  ,53.2 
46  31.3 
49  9.6 
51  48.3 
54  27.2 

57    6.3 

59  45J 

225^3 

5    5.1 

7  45.1 

10  25.2 
13  5  6 
15  46  0 
18  26.7 
21     7.5 

23  48.4! 
26  29.4 

29  10.5, 

31  51.7 

34  33.0J 

37  14.3] 

39  55.7 

42  37.1 
45  18  6 
48    0.0 

89° 


Diff: 

for  10 


.52.1 
.52.2 
52.3 
52.4 
52.5 
52.6 

.52.7 
52.8 
52.9 
53.0 
53.0 

.53.1 
53.2 
.53.3 
53.3 

53.4 

53.5 
.5-3.5 
.53.0 
53.6 

53.6 

.53.7 
.53.7 
.53.7 
,53.7 
53.8 

,53.8 
.53.8 
.53.8 
53.8 


Vis 


91° 


7  48.9 
10  24.7 
13  0.1 
15  3,5.1 
18  9.8 
20  44.0 

23  17.9 
25  51.2 
28  24.2 
30  56.7 
33  28.7 

.36  0.2 
38  31.3 
41  1.8 
43  31.9 
46    1.4 

48  30.4 
.50  .58.8 
53  26.6 
55  53.9 
58  20.7 


0  46  8 
3  12  3 
5  37.2 
8  1.5 
'10  25.2 

Il2  48.2 
15  10.5 
17  32.2 
19.53  1 
22  13.4 

J92° 

I    XIs 


DiH'. 

for  10 


51.9 
51.8 
51.7 
51.6 
51.4 

51.3 

51.1 
51.0 
50.8 
.50.7 
50.5 

,50.4 
50.2 
,50.0 
49.8 

49.7 

49.5 
493 
49.1 
48.9 

48.7 

48.5 
483 
48.2 
47.9 

47.7 

47.4 
47.2 
47.0 
46.7 


VIIs 


93= 


26  10  4 

28  43 

29  57.1 
31  49.0 
33  39.0 
35  29.7 

37  18.4 

39  6.2 

40  52.9 
42  38.4 
44  23.0 

46  6.5 

47  48.8 
49  30.1 

51  10.3 

52  49.4 

.54  27.3 
.')6  4.2 
.57  39.9 
.59  14.4 
T47.8 

2  20.1 

3  51  2 

5  21.1 

6  49.9 

8  17.4 

9  43.8 

11  9.0 

12  33  0 

13  55  8 
15  17.3 

94° 


Diff: 
for  10 


38.0 
37.6 
37.3 
37.0 
36.6 

36.2 

359 
35.6 
352 
34.9 
34  5 

34.1 
33.8 
33.4 
330 

32.6 

323 
31.9 
31  5 
31.1 
30.8 

30.4 
30.0 
29.6 
29.2 

28.8 

28.4 
28.0 
27.6 
27.2 


VIIIs 


94° 


46  11  3 

46  ,52.6 

47  32.5 

48  11.0 

48  48.1 

49  23.9 

49  58.2 

50  31.2 

51  2.9 

51  33.1 

52  1.9 

52  29.4 

52  55.4 

53  20. 1 

53  43.3 

54  5.2 

54  25  6 

54  44.6 
.55  2.3 
.55  18.5 

55  33.3 

55  46.8 
.55  58.8 
.56    9.4 

56  18.5 
56  26.3 

.56  32.7 
56  37.6 
.56  41.2 
.56  43.3 
56  44.0 

94° 

IXs 


15  0 
30 

14  0 
30 

13  0 
30 

12  0 
30 

11  0 
30 

10  0 

30 
9  0 

30 
8  0 

30 

7  0 
30 

6  0 
30 

5  0 

30 
4  0 

30 
3  0 

30 

2  0 
30 

1  0 
30 

0  0 


TABLE  LVI. 


81 


Equation  II  of  the  Maori's  Polar  Distance. 
Argument  11,  corrected. 


Ills    diff.    IVs      diff.     Vs      diff     Vis     diff.    VIIs      diff.    VIIIs    diff. 


0  13.8 
0  13.9 
0  14.1 
0  14.5 
0  15.1 
0  15.8 

0  16.7 
0  17.7 
0  18.9 
0  20.3 
0  21.8 

0  23.5 
0  25.3 
0  27.3 
0  29.4 
0  31.7 


0 
1 
2 

3 
4 
5 

6 

7 
8 
9 
10 

11 
12 
13 

14 
15 

16  0  34.2 

17  0  36.8 

18  jo  39.6 

19  0  42.5 

20  0  45.5 


0  48.7 
0  52.1 
0  55.6 

0  59.3 

1  3.1 

1  7.0 
1  11.1 
1  15.4 
1  19.8 
1  24.4 

II» 


0.1 
0.2 
0.4 
0.6 
0.7 

0.9 

1.0 
1.2 
1.4 
1.5 
1.7 

1.8 
2.0 
2.1 
2.3 

25 

2.6 
2.8 
2.9 
3.0 

3.2 

34 
35 
3.7 
3.8 
3.9 

4.1 
4.3 
4.4 
4.0 


1  24.4 
1  29.0 
1  33.8 
1  38.7 
1  43  8 
1  49.0 

1  54.3 

1  59.8 

2  5.4 
2  11.1 
2  16.9 

2  22.9 
2  29.0 
2  35.2 
2  41.5 
2  47.9 

2  54.5 

3  1.1 
3  7.9 
3  14.8 
321.8 

3  28  8 
3  36  0 
3  43  3 
3  50.7 

3  5S.2 

4  5.8 
4  13.4 
4  21.2 
4  29.0 

,4  36.9 

I» 


4  36.9 
4  44.9 

4  53.0 

5  1.1 
5  9.3 
5  17.6 

5  26.0 
5  34.4 
5  42.9 

5  51.4 

6  0.0 

6  8.7 
6  17.4 
6  26.2 
6  35.0 
6  43.8 

6  52.7 

7  1.6 
7  10.6 
7  19.6 
7  28.6 

7  37.7 
7  46.8 

7  55.9 

8  5.0 
8  14.1 

8  23.3 
Is  32.5 
,8  41.6 

8  50.8 
|9    0.0 


8.8 
8.9 

8.9 
9.0 
9.0 
9.0 

9.1 

9.1 
9.1 
!  9.1 
9.1 
9.2 

9.2 
9.1 
9.2 
9.2 


9  00 
9  9.2 
9  18.4 
9  27.5 
9  36.7 
9  45.9 

9  55.0 
10  4.1 
10  13.2 
10  22.3 
10  31.4 

10  40.4 
10  49.4 

10  58.4 

11  7.3 
11  16.2 

11  25.0 
11  33.8 
11  42.0 

11  51.3 

12  0.0 

12  8.6 
12  17.1 
12  25.6 
12  34.0 
12  42.4 

12  50.7 

12  58.9 

13  7.0 
13  15.1 
13  23.1 


XIs 


9.2 
9.2 
9.1 
92 
9.2 

9.1 

9.1 
9.1 
9.1 
9.1 
9.0 

9.0 
90 

8.9 
8.9 


13  23.1 
13  31.0 
13  38.8 
13  46.6 

13  54.2 

14  1.8 

14  9.3 
14  16.7 
14  24.0 
14  31.2 
14  38.2 

14  45.2 
14  52.1 

14  58.9 

15  5.5 
15  12.1 


18.5 
24.8 
31.0 
37.1 
43.1 

48.9 

54.6 

0.2 

5.7 

11.0 

16.2 
21.3 
26.2 
31.0 
35.6 


7.9 

7.8 
7.8 
7.6 
7.6 

7.5 

7.4 
7.3 
7.2 
7.0 
7.0 

6.9 
6.8 
C.6 
6.6 

6.4 

6.3 
6.2 
6.1 
6.0 

5.8 

5.7 
5.6 
5.5 
5.3 
5.2 

5.1 
4.9 

4.8 
4.6 


Xs 


16  35.6 
16  40.2 
16  44.6 
16  48.9 
16  53.0 

16  56.9 

17  0.7 

17  4.4 
17  7.9 
17  11.3 
17  14.5 

17  17,5 
17  20.4 
17  23.2 

17  25.8 
17  28.3 

17  30.6 
17  32.7 
17  34.7 
17  36.5 
17  38.2 

17  39.7 
17  41.1 
17  42.3 
17  43.3 

17  44.2 

17  44.9 
17  45.5 
17  45.9 
17  46.1 
17  46.2 

IXs 


4.6 
4.4 
4.3 
4.1 
3.9 

3.8 

3.7 
3.5 
3.4 
3.2 

3.0 

2.9 
2.8 
2.6 
2.5 

2.3 

2.1 
2.0 
1.8 
1.7 

1.5 

1.4 
1.2 
1.0 
0.9 

0.7 

0.6 
0.4 
0.2 
O.I 


30 
29 
28 
27 
26 
25 

24 
23 
22 
21 
20 

19 
18 
17 
16 

15 

14 
13 
12 
11 
10 

9 

8 
7 
6 
5 

4 
3 
2 
1 
0 


TABLE  LVIL 

Equation  HI  of  Moon's  Polar  Distance. 

Argument.     Moon's  True  Longitude. 


o 

Ills 

IVs 

Vs 

Vis 

VIIs 

VIIIs 

" 

" 

,' 

/' 

// 

/. 

o 

0 

16.0 

14.9 

12.0 

8.0 

4.0 

1.1 

30 

6 

16.0 

14.5 

11.3 

7.2 

33 

0.7 

24 

12 

15.8 

13.9 

10.5- 

6.3 

2.6 

0.4 

18 

18 

15.6 

13.4 

9.7 

5.5 

2.1 

0.2 

12 

24 

15.3 

12.7 

8.8 

4.7 

1.5 

00 

6 

30 

14.9 

12.0 

8.0 

4.0 

1.1 

0.0 

0 

lis 

Is 

0» 

XIs 

Xs 

IXs 

K 


€2TABLK  LVIII. 

To  convert  Degrees 
and  Minutes  into 
Decimal  Parts. 


TABLE  LIX. 

Equations  of  Maori's  Polar  Distance. 
Aigmiierit?,  Arg.  20  of  Long.;   V  U>  IX 

corrected;  X  not  corrected;  and  XI 

and  Xll  corrected. 


Deg 

(StMin. 

Dec  1 
parts. 

Arg  J 

20 

V 

VI 

VII 

vm 

IX  X 

XI 

Arg 

Arg 

XII 

) 

Arg, 

0  ' 

1  5 

003 

250 

0.3 

55.9 

6.1 

2,6  25.1 

3.0  0  7 

0.9 

250 

0 

4.0 

500 

1  26 

4 

260 

0.3 

55.8 

6.2 

2,7  25,1 

3.1!  0,7; 

0.9 

240 

10 

3,7 

510 

148 

5 

270 

0.4 

55.7 

6.3 

2,8  25.0 

3,2  0,8 

1.0 

230 

20 

3.4 

520 

2  10 

6 

280 

0.6 

55.4 

6.5 

3.0  24.9 

3,5 

1,0; 

1.0 

220 

30 

3,1 

530 

2  31 

7 

290 

0.8 

55.1 

6.9 

3.3  24.8 

3,8 

1,2 

1.1 

210 

■iV 

2,8 

540 

2  53 

8 

300 

1.0 

54.6 

7.3 

3.7  24.7 

4.3 

1.5 

1.2 

200 

50  j 

2,5 

55( 

3  14 

9 

310 

1.3 

54.1 

7.8 

4.2  24.4 

4.9 

1.8 

1.3 

190 

60 

2.3 

.501 

3  36 

10 

320 

1.7 

53.4 

8.4 

4.7  24.1 

5.6 

2,2 

1.4 

ISO 

70 

2,1 

.570 

3  58 

11 

330 

2.1 

52.7 

9.1 

5.4  23.8 

6,4 

2,7; 

1.5 

170 

80 

1,9 

5H0 

4  19 

12 

340 

2.6 

51.9 

9.8 

6.1,23.5 

7,2 

3,2 

1.7 

160 

90 

1.7 

590 

4  41 

13 

350 

3.1 

51.0 

10.7 

6.9'23,2 

8,2 

3.8 

1.9 

150 

100 

1.6 

600 

.5  2 

14 

360 

3.7 

50.0 

11.6 

7,7  22,8 

9.2 

4.4 

2.1 

140 

110 

1.5 

610 

5  24 

15 

370 

4.3 

48.9 

12.6 

8,7  22,4 

10,3 

5,1 

2.3 

130 

120 

1.5 

620 

5  46 

16 

380 

4.9 

47.7 

13.6 

9.7  21.9 

11.5 

5.8 

2.5 

120 

130 

1.5 

630 

6  7 

17 

390 

5.6 

46.5 

14.8 

10.7|21.4 

12,8 

6.6 

2.8 

11) 

140 

1.5 

640 

6  29 

18 

400 

6.4 

45.2 

16.0 

11,8  20,9 

14,1 

7,4 

3.0 

100 

150 

1,6 

650 

6  50 

19 

410 

7.1 

43.9 

17.2 

13,0  20,4 

15,5 

8.3 

3.3 

90 

160 

1.7 

660 

7  12 

20 

420 

7.9 

42.5 

18.5 

14,2  19,9 

17,0 

9.1 

3.5 

80 

170 

1.9 

670 

7  34 

21 

430 

8.8 

41.0 

19.8 

15.5  19.3 

18.5 

10.1 

3.8 

70 

180  2.1 

680 

7  55 

22 

440 

9.6 

39.5 

21.2 

16.8  18.7 

20.1 

11.0 

4.1 

60 

190  2,3 

690 

8  17 

23 

450 

10.5 

38.0 

22.6'l8.l'l8.1 

21.7 

12,0 

4,4 

50 

2002.5 

700 

8  38 

24 

460 

11.3 

36.4 

24.1 119. 4  17.5  23.3 

12.9 

4.7 

40 

210  2  8 

710 

9  0 

25 

470 

12.2 

34.9 

25.5;20.8  16.9124.9 

13,9 

5.11 

30 

2203  1 

720 

9  22 

26 

480 

13.3 

3S.2 

27.0|22.2  16.3  26.6 

15,0 

5,4 

20 

230 

3.4 

730 

9  43 

27 

490 

14.1 

31.6 

28.5,23,6  15.6  28.3 

16.0 

5,7 

10 

240 

3.7 

740 

10  5 

28 

.500 

15  0 

30.0 

30,0  25  0  15.0  30.0 

17,0 

6.0 

0 

250 

4,0 

750 

10  26 

29 

510 

15  9 

28  4 

31.5 

'26.4  14.4 

31.7 

18.0 

6.3 

990 

260 

4,3 

760 

10  48 

30 

520 

16.8 

26.8 

33.027. 8  13.7 

33.4 

19.0 

6.6 

9S0 

270 

4,0 

770 

11  10 

31 

530 

17.8 

25.1 

34,5  29.2  13.1 

35.1 

20.1 

7.0 

970 

280 

4,9 

780 

1131 

32 

540 

18.7 

23,6 

35,9.30.6  12.5 

36,7 

21.1 

7.3 

960 

290 

5.2 

790 

1153 

33 

550 

19.5 

220 

37,4  31,9  11.9 

38.3 

22.0 

7.6 

9.50 

300 

,5.5 

800 

12  14 

34 

5fi0 

20.4 

20.5 

38  8  33.2  11,3  39,9 

23.0 

7,9 

940 

'310 

,5.7 

810 

12  36 

35 

570 

21.2 

19.0 

40,2  34,5  10.7  41.5 

23.9 

8,2 

930 

.320 

5,9 

820 

12  58 

36 

580 

22.1 

17.5 

41,5  35,8  10,1 

43.0 

24.9 

8,5 

920 

330 

;6,i 

830 

13  19 

37 

590 

22.9 

16.1 

42,8  37,0:  9,6 

1 

44,5 

25.7 

8,7 

910 

340 

6.3 

840 

13  41 

38 

600 

23.6 

14.8 

44,0  38.2'  9,1 

45.9 

26.6 

9,0 

900 

3>0 

64 

850 

14  2 

39 

610 

24.4 

13.5 

45  2  39,3  8,6 

47.2!27.4 

92 

890 

360 

6.6 

860 

14  24 

40 

620 

25.1 

12.3 

46.4  40.3 

8.1 

48,528,2 

9,5 

880 

370 

6.5 

870 

14  46 

41 

630 

25.7 

11.1 

47.4  41,3 

7.6[49.7  28.9 

9,7 

870 

380 

6.5 

880 

16  7 

42 

640  26.3 

10.0 

48,4  42,3'  7,2  50,8  29,6 

1                     1 

9,9 

850 

3'.)0 

6.5 

890 

15  29 

43 

650 '26.9 

9.0 

49,3  43,1!  6,8  51,8  30.2 

10,1 

850 

400 

'6.4 

900 

15  50 

44 

660  27.4 

8.1 

50.2  43.9  6,552,8  30,8 

10.3 

840 

410 

!6.3 

910 

16  12 

45 

670  27.9 

7.3 

50,9  44.6,  6, 253.6  31,3 

10.5  830 

420 

6.1 

920 

16  34 

46 

680  28.3 

6.6 

51.6  45  3  5,9  54.4  31.8 

10.6  820 

430 

5.9 

930 

16  55 

47 

690 

28.7 

5.9 

52.2  45,8,  5,6  55.132.2 

10.7|810 

440 

5.7 

940 

17  17 

48 

700 

29.0 

5.4 

52,7  46,3  5,3  55,7  32.5 

10.8  800 

450 

5  5 

y.=o 

17  38 

49 

710 

29.2 

4.9;53.1  46.7  5.2  56.2  32.8 

10.9  790 

460 

5.2 

!60 

18  0 

50 

720 

29.4 

4.6  53.5  47.0  5.1  56.5  33.0 

11.0  780 

470 

4.9 

970 

18  22 

51 

730 '29.6 

4.3:53.747.2  5.0  56.8  33.2 

11,0  770 

480 

4.6 

980 

18  43 

62 

740  29.7 

4.253.8  47.3  4.9  56.9  33.3 

11.1  760 

490 

4.3 

990 

19  5 

63 

750  29.7 

4.1I53.947.4  4.9,57.0  33.3  11. li  750 

500 

4  0  1000 

Co 

QStac 

t8 

TABLE  LX.  TABLE  LXI.         a3 

Small  Equations  of  Moon's  Parallax.       Moon's  Equatorial  Parallax. 
Args.,  1,  2,  4,  5,  6,  8, 9,  12,  13,  of  Long.         Argument.  Arg.  of  Evection. 


A.    1 

2 

4 

5 

6 

8 

9 

12 

13 

0  0.0 

l.G 

0.6 

1.0 

1.9 

0.0 

3.1) 

1.4 

2.0 

3  0.0 

1.6 

0.6 

1.6  1.9 

0.0 

3.5 

1.4 

2.0 

0  0.0 

1.5 

0.6 

1.5  1.8 

0.0 

3.1 

1.4 

1.9 

0  0.1 

1.5 

O.fi 

1.5  1.8 

0.1 

2.6 

1.3 

1.8 

12  0.1 

1.4 

0.5 

1.4  1.7 

0.2 

1.9 

1.2 

1.7 

15  0.1 

1.3 

0.5 

1.3  1.6 

0.2 

1.3 

1.1 

1.6 

18  0.2 

1.1 

0.4 

1.1   1.4 

03 

0.7 

1.0 

1.4 

■>1  0  3 

1.0 

0.4 

1.0  1.3 

05 

02 

0.9 

1.2 

24  0.4 

0.9 

0.3 

0.9  1.2 

0.6 

0.0 

0.7 

1.0 

27  0  .5 

0.7 

0.3 

0.7  1.0 

0.7 

0.1 

06 

0.9 

-.50  0.5 

0.6 

02 

0.6  0.9 

0.8 

0.4 

0.5 

0.7 

.J3  0.6 

0.4 

0.2 

0.4  0.7 

0.9 

OS 

0.4 

0.5 

Sfi  0  7 

0.3 

01 

0.3  0  6 

1.0 

1.5 

03 

0.4 

39  0  7 

02 

0.1 

0.2  0.5 

1.1 

2.1  0.2 

0.2 

42  0.8 

0.1 

0.0 

0.1  0.4 

1.1 

2.8  0.1 

0.1 

15  OS 

0.0 

00 

0.0  0.3 

1.2 

3.2  0.0 

0.0 

48  0.8 

0.0 

0.0 

0.0  0.3 

1.2 

3.5  0.0 

0.0 

50  OS 

0.0 

0.0 

0.0 

03 

1.2 

3.6 

0.0 

0.0 

100 
97 
94 

91 

88 
85 

82 
79 
76 

73 
70 
67 

64 
61 

58 

55 
52 
50 


Constant    7" 

The  first  two  figures  only  of  the  Arguments 
arc  taken. 


0 

0 

0. 

Is 

11* 

Ills 

42.6 

IVs 
24.1 

10.8 

o 
30 

120.8 

1  1.5.6 

1    1.5 

1 

1  20.8|l 

15.2 

1    0.9 

41.9 

23.6 

10.5 

29 

2 

1  20.8  1 

14.9 

1    0.3 

41.3 

23.0 

10.2 

28 

3 

1  20.7il 

14.5 

59.7 

40.6 

22.5 

9.9 

27 

4 

1  20.7,1 

14.2 

59.2 

40.0 

21.9 

9.6 

26 

5 

1  20.6  1 

1 

13.8 

58.6 

39.4 

21.4 

9.4 

25 

6 

1  20.6  1 

13.4 

57.9 

38.7 

20.9 

9.1 

24 

7 

1  20.5  1 

13.0 

57.3 

38.1 

20.4 

8.8 

23 

8 

1  20.4  1 

12.6 

56.7 

37.4 

19.9 

8.6 

22 

9 

1  20.3  1 

12.2 

56.1 

36.8 

19.4 

8.4 

21 

10 

1  20.2  1 

1 

11.7 

55.5 

36.1 

18.9 

8.2 

20 

11 

1  20.1  1 

11.3 

54.9 

35.5 

18.4 

8.0 

19 

12 

1  19.9  1  10.8 

54.2 

34.9 

17.9 

7.8 

18 

13 

1  19.8  1 

10.4 

53.6 

34.2 

17.5 

7.6 

17 

14 

1  19.6  1 

9.9 

53.0 

33.6 

17.0 

7.4 

16 

15 

1  19.5  1 

1 

9.4 

52.3 

33.0 

16.6 

7.2 

15 

16 

1  19.3  1 

9.0 

51.7 

32.4 

16.1 

7.1 

14 

17 

1  19.1  1 

8.5 

51.1 

31.7 

15.7 

6.9 

13 

18 

1  18.9  1 

8.0 

50.4 

31.1 

15.2 

6.8 

12 

19 

1  18.7  1 

7.5 

49.8 

30.5 

14.8 

6.7 

11 

20 

1  18.4  1 

7.0 

49.1 

29.9 

14.4 

6.5 

10 

21 

1  18. 21 

6.5 

48.5 

29.3 

14.0 

6.4 

9 

22 

I  18.0  1 

5.9 

47.8 

28.7 

13.6 

6.3 

8 

23 

1  17.7  1 

5.4 

47.2 

28.1 

13.2 

6.3 

7 

24 

1  17.4  1 

4.8 

46.5 

27.5 

12.9 

6.2 

G 

25 

117.1 

1 

4.3 

45.9 

26.9 

12.5 

6.1 

5 

26  1  16.9 

1 

3.8 

45.2 

26.3 

12.1 

6.1 

4 

27  1  16.6 

1 

3.2 

44.6 

25.8 

11.8 

6.1 

3 

28  1  16.2 

1 

2.6 

43.9 

25.2  11.5 

6.0 

2 

29  I  15.9  1 

2.1 

43.3i24.7lll.l 

60 

1 

30  1  15.6;i 

1.5 

42.6|24.l|l0.8 

6.0 

0 

XIs 

Xs 

IX* 

VIIIs 

vii»|vi» 

84 


TABLE  LXII. 

MoorCs  Equatorial  Parallax. 

Argument.      Anomaly. 


Os 


diff 


diff 


lis 


diff 


Ills    diff 


IV« 


diff  t     V»     diff 


0 
1 
2 
3 
4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 


58  57.7' 
58  57.7 
58  57.6 
58  57.4 
58  57.1 
53  56.8 

58  56  4 
58  56.0 
58  55.4 
58  54.8 
58  54.2 

58  53.4' 
58  52.6 
58  51.8 
58  50.8 
58  49.8 


58  27.0 
"■"  58  25.0 
"J  58  23.0 
y  58  20.9 
"•^  58  18.7 
"•"^  58  16.5 
|0.4 

58  14.3 
58  12.0 


0.6 
0.6 
0.6 

10.8 

0.8 
0.8 
1.0 
1.0 
1.1 


58  48.7 
58  47.6 


1.1 

ll  2 
53  46.4,  X 

58  45.l|:'o 
58  43.8i 

1.4 


oS 
53 
58 

58 

57  59.8 
57  57.2 
57  54.6 
5751.9 


2.0 
2.0 
2.1 

2.2 
2.2 

2.2 

23 

9  6  2-* 
70  2.4 

4:82-* 

2.5 
2.3 


57  49.2 
57  46.4 
57  43.7 
57  40.8 
57  33.0 


57  35  1 
57  32.2 
57  29.3 


58  42.4',  = 

58  40.91,  = 

5S39.4,(, 

53  37.8  ifi  157  26.3 

53  36.2        j57  23.3 

1.8 
58  34.4 1,  7 '57  20.2 
58  32.7  ^■"— "'" 
5830.9| 


1.9 


57  17.2 
57  14.1 


2.5 
2.6 
2.6 
2.7 

12.7 

2.8 
2.7 
2.9 
2.8 

2.9 

2.9 
2.9 
30 
3.0 

3.0 

30 
3.1 
3  1 


58  29.0  2  0  5711.0^^ 


58  27.0 

Xl5 


57   7.9 


Xs 


'57  7.9 
57  4.8 
57  1.6 
|56  53.4 
56  55.2 
|56  52.0 

i56  48.8 
50  45.5 
56  42.3 
56  39.0 
56  35.7 

.56  32.4 
56  29.1 
.56  25.8 
56  22.5 
.56  19.2 

.56  15.9 
.56  12.6 
56    9.3 

55  6.0 

56  2.7 

.55  59.3 
55  56.0 
55  .52.7 
55  49.4 
55  46.1 

55  42  8 
i55  39.fi 
55  36.4 
55  33.1 
55  29.8 

IX» 


3.1 
3.2 
3.2 
3.2 
3.2 

32 

3.3 
3.2 
3.3 
3.3 

3.3 

3.3 
3.3 
33 
3.3 

3.3 

3.3 
33 
3.3 
3.3 

3.4 

33 
33 
33 
3.3 

3.3 

3.2 
3.2 
3.3 
3.3 


.55  29.8 
55  26.6 
55  23.4 
55  20.2 
55  17.0 
5513.8 

.55  10.6 
55  7.5 
55  4.4 
55  1.3 
.54  58.2 

54  55.1 
54  52.1 
H49.1 
54  46.1 
.54  43.1 

5440.2 
5437.3 
54  31.4 
54  31.5 
5428.7 

54  25.9 
54  23.1 
54  20.3 
5417.6 
5414.9 

54  12  2 
.54  9.6 
51  7.0 
54  4.4 
54    1.9 

VIIIs 


3.2 
32 
32 
3.2 
3.2 

3.2 

3.1 
3.1 
3.1 
3.1 

3.1 

3.0 
3.0 
3.0 
30 
2.9 

2.9 
2.9 
2.9 

2.8 

2.8 

2.8 
2.8 
2.7 
2.7 
2.7 

2.6 
2.6 
2.6 
2.5 


.54  1.9 
53  59.4 
.53  55.9 
53  54.5 
.53  52.1 
.53  49.7 

.53  47.4 
53  45.1 
53  42.9 
.53  40  6 
.53  38.5 

.53  36.3 
53  34.2 
53  32. 1 
53  30.1 
53  28.1 

53  26.2 
53  24.3 
53  22  4 
.53  20.6 
.53  18.8 

53  17.0 
53  15.3 
53  13.7 
.53  12.0 
53  10.4 


2.5 

2.5 
2.4 
2.4 
2.4 
2.3 

2.3 

2.2 
2.3 
2.1 

2.2 

2.1 
2.1 
2.0 
2.0 

1.9 

1.9 
1.9 
1.8 
1.8 
1.8 

1.7 
1.6 
1.7 
1.6 
1.5 

1.5 

1.5 
1.4 
13 


VIIs 


53  3.2 
53  1.8 
53  0.5 
•52  59  3 
.52  58.1 
52  57.0| 
1 
.52  55.8 
52  54  8 
.52  53.3, 
.52  52.8 
52  51.9 

.52  51.0 
.52  50.1 
.52  49.3 
52  48.6 
52  47.9 

52  47.2 
52  46.6 
52  46.0 
.52  45.5 
52  45.0 

52  44.6 
.52  44.? 
52  43  8 
.52  43.5 
52  43.3 

52  43.1 
52  42.9 
52  42  8 
52  42.7 
.52  42.7 

I    Vis 


1.4 
13 
1.2 
1.2 
1.1 

1.2 

1.0 
1.0 
1.0 
0.9 

0.9 

09 
0.8 
0.7 
0.7 

0.7 

0.6 
06 
0.5 
0.5 

0.4 

0.4 
0.4 
03 

0.2 

0.2 

02 
0.1 
0.1 
0.0 


30 

29 
28 
27 
26 
25 

24 
23 
22 
21 
20 

19 
18 
17 
16 
15 

14 
13 
12 
11 
10 

9 

8 
7 
6 
5 

4 
3 
2 
1 
0 


TABLE    LXIII. 


85 


MoorCs  Equatorial  Paj-allax, 
Argument.     Argument  of  the  Variation. 


Os 

Is 

II* 

Ills 

IV* 

V* 

o 
0 

" 
55.6 

42.3 

16.0 

3.7 

17.6 

44.0 

0 

30 

1 

55.6 

41.5 

15.3 

3.8 

18.5 

44.8 

20 

2 

55.5 

40.7 

14.5 

3.8 

19.3 

45.6 

23 

3 

55.5 

39.8 

13.8 

3.9 

20.1 

463 

27 

4 

553 

39.0 

13.1 

4.1 

21.0 

47.0 

26 

5 

55.2 

38.1 

12.4 

4.3 

21.9 

47.7 

25 

6 

55.0 

37.2 

11.7 

4.5 

22.7 

48.4 

24 

7 

54.8 

3S.3 

11.1 

4.7 

23.6 

49.1 

23 

8 

54.6 

35.5 

10.4 

5.0 

24.5 

49.7 

22 

9 

54.3 

34.6 

9.8 

5.3 

25.4 

50.3 

21 

10 

54.0 

33.7 

9.2 

5.6 

26.3 

50.9 

20 

11 

53.7 

32.7 

8.7 

6.0 

27.2 

51.5 

19 

12 

53.3 

31.8 

8.2 

6.3 

23.2 

52.1 

18 

13 

52.9 

30.9 

7.7 

6.8 

2J.1 

52.6 

17 

14 

52.5 

30.0 

7.2 

7.2 

30.0 

53.1 

16 

15 

52.0 

29.1 

6.7 

7.7 

30.9 

53.5 

15 

16 

51.5 

28.2 

6.3 

8.2 

31.8 

54.0 

14 

17 

51.0 

27.2 

5.9 

8.7 

32.8 

54.4 

13 

18 

50.5 

26.3 

5.6 

9.3 

33.7 

54.8 

12 

19 

49.9 

25.4 

5.3 

9.8 

34.6 

55.1 

11 

20 

49.4 

24.5 

5.0 

10.5 

35.5 

55.4 

10 

21 

48.8 

23.6 

4.7 

11.1 

36.4 

55.7 

9 

22 

48.1 

22.7 

4.5 

11.7 

37.3 

56.0 

8 

23 

47.4 

21.9 

4.3 

12.4 

38.2 

56.2 

7 

24 

46.8 

21.0 

4.1 

13.1 

39.0 

56.4 

6 

25 

46.1 

20.1 

3.9 

13.8 

39.9 

56.6 

5 

26 

45.4 

19.3 

3.8 

14.5 

40.8 

56.8 

4 

27 

44.6 

18.5 

3.7 

15.3 

41.6 

56.9 

3 

28 

43.9 

17.6 

3.7 

16.1 

42.4 

56.9 

2 

29 

43.1 

16.8 

3.7 

16.8 

43.2 

57.0 

1 

30 

42.3 

16.0 

3.7 

17.6 

44.0 

57.0 

0 

XI» 

X. 

IX* 

VIII* 

VII* 

VI* 

86        TABLE  LXIV. 


TABLE  LXV. 


Reduction  of  the  Parallax, 
and  also  of  the  Latitude. 

Arffiiinent.     Latitude. 


Aloon^s  Semi-diameter. 
Argument.     Equ;itorial  Parallax. 


Lat. 

iRed. 

Red.  of 
Lat. 

% 

Par  Se 

nidia. 

Eq.Par  Se 

midia 

Ei.Par 

Sem'dia.  sec  fro. 
Par. 

'"''■"' 

"i     ' 

/     ■' 

, 

,, 

/    // 

1 

0 

" 

,       „ 

53 

0    14 

265 

56     0 

15 

15.6 

59     0 

16     46 

I'o.a 

0 

00 

0     0.0 

53 

10    14 

29.3 

56  10 

15 

18.3 

j59    10 

16     7.4 

2    0.5 

3 

0.0 

1    11.8 

53 

20    14 

320 

56  20 

15 

21.0 

59  20 

16  10  1 

3   0.8 

6 

0.1 

2  22  7 

53 

30    14 

34.7 

56  30 

15 

23  8 

59  30 

16   128 

4    1.1 

9 

03 

3  32  1 

53 

40  j  14 

37.4 

56  40 

15 

26.5 

59  40 

16   15.6 

5    1.4 

12 
15 

0.6 
0.7 

4  30.3 

5  43.4 

53 

50    14 

40.2 

56  50 

15 

292 

59  50 

16   18.3 

6    1.6 

54 

0 

14 

42.9 

57     0il5 

31.9 

6'J     0 

16  21.0 

7    1.9 

18 

1.0 

6  43  7 

54 

10 

14 

456 

57  10 

15 

34.7 

60   10 

16  23.7 

8    2.2 

21 

1.4 

7  39  7 

54 

20 

14 

48.3 

57  20 

15 

37.4 

60  20 

16  2(3.4 

9    2.4 

24 

1.8 

8  30.7 

54 

30 

14 

51.1 

57  30 

15 

40.1 

CO  30 

16  29.2    10  12.7 

27 
3.) 

2.3 

2.7 

9  16.1 
9  55.4 

54 

40 

14 

53.8 

57  40 

15 

428 

60  40 

16  31  9 

54 

50 

14 

56.5 

57  50 

15 

45.6 

60  50 

16  34.6 

33 

3.3 

10  28  3 

5.5 

0 

14 

59.2 

58     0 

15 

48.3 

61      0 

16  37.3 

36 

3.8 

10  54.3 

55 

10 

15 

2.0 

58   10 

15 

51.0 

61    10 

16  40.1 

39 

4.4 

11   13.2 

55 

20 

15 

4.7 

58  20 

15 

53.7 

61   20 

16  42.8 

42 
45 

4.9 
5.5 

11  21.7 
11  28.7 

55 

30 

15 

7.4 

58  30 

15 

56.5 

61   30 

16  45.5 

55 

40  1  15 

10.1 

58  40 

15 

59.2 

|61   40 

16  43  2 

48 

6.1 

11   2.-).2 

55 

50  1  15 

12.9 

53  50 

10 

1.9 

61   50 

16  51.0 

51 

6.7 

11    14.1 

56 

0ll5 

15.6 

59   0  Ue 

4.6 

62     0 

16  53.7 

54 

7.2 

10  5.5.7 

57 

7.8 

10  31.0 

60 

83 

9  57.4 

63 

8.8 

9  18.3 

66 

9.2 

8  32.9 

69 

9.7 

7  42.0 

TABLE  LVL 

72 

10.0 

6  4.1.9 

75 

10.3 

5  45.4 

A  u^m 

entation  of  Moun's  Semi-diameter.. 

78 

10.6 

4  41.0 

81 
84 

10.8 
11.0 

3  33.5 
2  23.7 

Horizon.  Semi-diameter. 

i 

Horizon.    Semi-diameter. 

87 

11.1 

1   123 

Alt. 



Alt. 

90 

11.1 

0     00 

14'30" 

15' 

16 

17 

1 

14'  30" 

15' 

16 

r7 

Subsidiary  Table. 

o 
2 
4 

0.6 
1.0 

0.0 
1.1 

07 
1.3 

0.8 
1.5 

1    ° 
42 
45 

9.2 

9.7 

9.8 
10.4 

11.2 
11.8 

12.6 
133 

Lat. 

+  3' 

—  3' 

o 

6 

8 

1.5 
2.0 

1.6 

2.1 

1.9 
2.4 

2.1 
2.7 

;48 
51 

10.2 
10.6 

10.9 
11.4 

12.4 
13.0 

14.0 
14.7 

" 

" 

0 

+  0.0 

—  0.0 
0.0 
0.0 

10 

2.4 

2.6 

3.0 

3.4 

54 

11.1 

11.8 

13.5 

15.2 

6 
12 
15 
18 
24 

0.0 
0.0 
0.0 

12 

2.9 

3.1 

3.6 

4.0 

57 

11.5 

12.3 

14.0 

15.8 

0.0 

14 

3.4 

3.6 

4.1 

4.7 

60 

11  8 

12.7 

14.4 

16.3 

O.I 

0.1 

16 

3.8 

4.1 

4.7 

5.3 

63 

12.2 

13.0 

14.9 

16.8 

0.1 

0.1 

18 

4.3 

4.6 

5.2 

5.9 

66 

12.5 

13.4 

15.2 

17.2 

0.1 
0.2 

21 

4.9 

5.3 

6.0 

6.8 

69 

12.8 

13.7 

15.6 

17.6 

30 

36 
42 
48 

0.1 
0.2 
0.2 
0.3 

24 

5.6 

6.0 

6.8 

7.7 

72 

130 

13.9 

15.9 

17.9 

0.2 

27 

6.2 

6.7 

7.6 

8.6 

75 

13.2 

14.1 

16.1 

18.2 

0.3 

30 

6.9 

7.3 

8.4 

9.5 

78 

13.4 

14.3 

16.3 

18.4 

64 

0.3 

0.3 

33 

7.5 

8.0 

9.1 

10.3 

81 

13.5 

14.4 

lfi.5 

18.6 

36 

8.1 

8.6 

9.8 

11.1 

84 

13.0 

14.5 

16.6 

18.7 

60 

0.4 

0.4 
0.5 

39 

8.6 

9.2 

10.5 

11.9 

90 

13.7 

14.6 

16.7 

18.8 

72 

0.5 

78 

0.6 

0.6 

84 

0.6 

0.6 

90 

+  0.6 

—  06 

TABLE   LXVII. 


87 


Mooii's  Horary  Motion  in  Longitude. 
Arsiiments.   1   to   18  of  Longitude. 


Arg. 

2 

0 

5.0 

2 

5.0 

4 

4.9 

6 

4.8 

8 

4.7 

10 

4.5 

12 

4.3 

14 

4.1 

IG 

3.8 

18 

3.6 

20 

3.3 

22 

30 

24 

2.7 

26 

2.3 

28 

2.0 

30 

1.7 

32 

1.4 

34 

1.2 

36 

09 

38 

0.7 

40 

0.5 

42 

0.3 

44 

0.2 

46 

0.1 

48 

0.0 

50 

0.0 

3 

4 

0.0 

2.9 

0.0 

2.8 

0.0 

2.8 

0.1 

2.8 

02 

2.7 

0.3 

2.6 

0.4 

2.5 

0.6 

2.3 

0.7 

2.2 

0.9 

2.0 

1.1 

1.9 

1.3 

1.7 

1.5 

1.5 

1.7 

1.3 

1.9 

1.2 

2.1 

1.0 

2.2 

0.8 

2.4 

0.7 

2.6 

0.5 

2.7 

0.4 

2.8 

0.3 

2.9 

02 

3.0 

0.1 

3.1 

0.0 

3.1 

0.0 

3.1 

0.0 

5 

6  i 

1 

7 

8 

1.9 

0.0 

0.00 

0.00 

0.00 

1.9 

0.0 

0.00 

0.00 

0.00 

1.9 

0.0 

0.01 

0.00 

0.02 

1.9 

0.1 

0.03 

0.01 

0.05 

1.8 

0.1 

0.06 

0.01 

0.09 

1.7 

0.2 

0.09 

0.02 

0.14 

1.7 

0.2 

0.13 

0.02 

0.19 

1.6 

0.3 

0.18' 

0.03 

0.26 

1.5 

0.4 

0.23 

0.04 

033 

1.4 

0.5 

0.28 

0.05 

0.41 

1.3 

0.6 

0.34 

0.06 

0.50 

1.1 

0.7 

0.40 

0.07 

0.58 

1.0 

0.8 

0.46 

0.08 

0.67 

0.9 

0.9 

0.52 

O.IO 

077 

0.8 

1.0 

0.58 

0.11 

0.8G 

0.7 

1.1 

0.63 

0.12 

0.94 

0.5 

1.2 

0.69 

0.13 

1.03 

0.4 

1.3 

0.74 

0.14 

1.11 

03 

1.3 

0.78 

0.15 

1.18 

03 

1.4 

0.82 

0.16 

1.25 

0.2 

1.5 

0.86 

0.16 

1.30 

0.1 

1.5 

0.89 

0.17 

1.35 

0.1 

1.6 

0.91 

0.17 

1.39 

00 

1.6 

093 

0.18 

1.42 

0.0 

1.6 

0.94 

0.18 

1.44 

0.0 

1.6 

0.94 

0.18 

1.44 

0.10 
0.15 
0.15 
0.14 
0.12 
0.10 

0.09 
0.07 
0.05 
0.03 
0.02 

0.01 
0.00 
0.00 
0.00 
0.01 

0.01 
0.03 
0.05 
0.06 
0.08 

0.10 
0.11 
0.12 
0.13 
0.13 


100 
98 
96 
94 
92 
90 

88 
86 
84 
82 
80 

78 
76 
74 
72 
70 

68 
66 
64 
62 
60 

58 
56 
54 
52 
50 


Arg. 

10 

11 

0 

0.00 

0.26 

2 

0.00 

0.25 

4 

0.02 

0.24 

6 

0.04 

0.22 

8 

0.08 

0.20 

10 

0.12 

0.17 

12 

0.16 

0.14 

14 

0.20 

0.11 

16 

0.24 

0.08 

18 

028 

0  05 

20 

0.31 

0.03 

22 

0.34 

0.01 

24 

0.35 

0.00 

26 

0.36 

0.00 

28 

0  35 

0.01 

30 

0.34 

0.02 

32 

0.32 

0.04 

34 

0.29 

0.06 

36 

0.26 

0.09 

38 

0.22 

0.11 

40 

0.18 

0.14 

42 

0.15 

016 

44 

0.12 

0.19 

46 

0.10 

0.21 

48 

0.09 

0.22 

60 

0.08 

022 

12 


0  00 
0.00 
0.01 
0.03 
0.04 
0.07 

0.09 
0.12 
0.16 
0.19 
0.23 

027 
031 
0  35 
039 
0.43 

0.47 
0.50 
0.54 
0.57 
0.59 

0.62 
0.63 
0.65 
0.66 
0.66 


13 

14 

0  00 

0.00 

0  00 

0.00 

000 

0.01 

0.01 

0.02 

0  02 

0.04 

0.03 

0,06 

0.04 

0.09 

0.00 

0.12 

0.07 

0.15 

0.09 

0.19 

0.11 

0.22 

0.13 

0.26 

0.15 

0.30 

0.17 

0.34 

0.19 

0.3S 

0.21 

0.42 

023 

0.45 

025 

0.49 

0.26 

0.52 

0.28 

0.55 

0.29 

0.58 

030 

0.60 

031 

0.62 

0.32 

0.63 

0.32 

0.64 

0.32 

0.64 

15 


000 
000 
000 
0.01 
001 
0.03 

0,02 
003 
004 
005 
O.OG 

007 
0  0>! 
0  08 
009 
0.10 

0.11 
0.12 
0.13 
0.14 
0.14 

0.15 
0.15 
0.16 
0.16 
0.16 


16 

17 

18 

0.26 

0.00 

0.21 

0  26 

0.00 

020 

0.2f) 

0.00 

0.20 

0.25 

0.00 

0.20 

0  25 

0.01 

020 

0.24 

O.OI 

0.20 

0.22 

0.02 

0.19 

0.21 

0.02 

0  19 

0.20 

0.03 

0  18 

0.10 

0  04 

0.18 

0.17 

0.05 

0.17 

0.15 

O.OG 

0.17 

0.14 

0.07 

0.16 

0.12 

0.07 

0.16 

0.11 

0.08 

0.15 

0.09 

0.09 

0.15 

0.07 

0.10 

0.14 

006 

O.ll 

0.14 

0.05 

0.12 

0.13 

0.04 

0.12 

0  13 

0.02 

0.13 

0.12 

0.01 

0.13 

0.12 

0.01 

0.14 

0.12 

0.00 

0.14 

0.12 

0.00 

0.14 

0  12 

000 

0.14 

Oil 

100 
98 
96 
94 
92 
90 

88 
86 
84 
82 
80 

78 
76 
74 
72 
70 

68 
66 
64 
62 
60 

58 
56 
54 
52 
50 


83  TABLE  LXVIII. 

Mooii's  Horary  Motion  in  Longitude. 
Artriimeiit.     Arjiiimeiit  of  the  Eveclion. 


Os 

Is 

II» 

III« 

IV  s 

Vs 

0 

0 

90.3 

74.7 

59.6 

39.4 

198 

5.9 

o 
30 

1 

80.3 

74.3 

5S.9 

38.7 

193 

5.6 

29 

2 

80.3 

73.9 

58.3 

33.0 

137 

5.3 

28 

3 

80.2 

73.5 

57.7 

37.3 

18.1 

5.0 

27 

4 

80.2 

73.1 

57.1 

36.6 

17.6 

4.7 

26 

5 

80.1 

72.7 

56.4 

36.0 

17.0 

4.4 

25 

6 

80.1 

72.3 

55.8 

35.3 

16.5 

4.1 

24 

7 

80.0 

71.9 

55.1 

346 

15.9 

3.8 

23 

8 

79.9 

71.4 

54.5 

33.9 

15.4 

3.6 

22 

9 

79.8 

71.0 

53.8 

33.2 

14.9 

3.4 

21 

10 

79.7 

70.5 

53.1 

32.5 

14.4 

3.1 

20 

11 

79.5 

70.1 

52.5 

31.9 

13.9 

2.9 

19 

12 

79.4 

69.6 

51.8 

31.2 

13.4 

2.7 

18 

13 

792 

69.1 

51.1 

30.5 

12.9 

2.5 

17 

14 

79  1 

63.6 

50.5 

29.9 

12.4 

2.3 

16 

15 

7S.9 

68.1 

49.8 

29.2 

11.9 

2.1 

15 

16 

78  7 

67.6 

49.1 

28.6- 

11.4 

2.0 

14 

17 

78.5 

67.0 

48,4 

27.9 

11.0 

1.8 

13 

18 

78.2 

66.5 

47.7 

27.2 

10.5 

1.7 

12 

19 

78.0 

66.0 

47.0 

26.6 

10.1 

1.6 

11 

20 

77.8 

65.4 

46.4 

26.0 

9.7 

1.4 

10 

21 

77.5 

64.9 

45.7 

25.3 

9.3 

1.3 

9 

22 

77.2 

64.3 

45.0 

24.7 

8.8 

1.2 

8 

23 

77.0 

63.7 

44  3 

24.1 

8.4 

1.2 

7 

24 

76.7 

63.2 

43.6 

23  5 

8.0 

1.1 

6 

25 

76.4 

62.6 

42.9 

22.8 

7.7 

1.0 

5 

2S 

76.1 

62.0 

42.2 

22.2 

7.3 

1.0 

4 

27 

75.7 

61.4 

41.5 

21.6 

6.9 

0.9 

3 

28 

75.4 

60.8 

40.8 

21.0 

6.6 

0.9 

2 

29 

75  0 

60.2 

40.1 

20.4 

6.2 

09 

1 

30 

74.7 

59.6 

39.4 

198 

5.9 

0.9 

0 

XIs 

X» 

IXs 

VIII* 

VIIs 

Vis 

Arguments. 


TABLE  LXIX. 
Moon's  Horary  Motion  in  Longitude. 
Sum  of  Equations,  2,  3,  &c.,  and  Eveclion  corrected. 


0" 

10" 

20" 

s 

o 

s     ° 

0 

0 

00 

0.2 

0.5 

XII    0 

I 

0 

0.0 

0.2 

0.4 

XI      0 

11 

0 

0.1 

0.2 

0.3 

X        0 

III 

0 

0.2 

02 

0.2 

IX      0 

IV 

0 

03 

02 

0.1 

VIII    0 

V 

0 

04 

0.2 

0.0 

VII     0 

VI 

0 

0.5 

0.2 

00 

VI      0 

1 

0" 

10" 

20" 

1 

TABLE   LXX. 


89 


Mooii's  Horary  Motion  in  Longitude. 
Arguments.     Sum  of  preceding  equations,  and  Anomaly  corrected. 


0" 
4.1 

10" 

20" 
6.5 

30" 
7,6 

40" 

8.8 

50" 
10.0 

60" 

70" 

12.4 

80" 
13.5 

90" 

100" 



0      0 

5.3 

11.2 

14.7 

15.9 

XII    0 

5 

4.1 

53 

6.5 

7.7 

8.8 

10.0 

11.2 

13.3 

13.5 

14.7 

15.9 

25 

10 

4.2 

5.4 

65 

7.7 

8.8 

10.0 

11.2 

12.3 

13.5 

146 

15.8 

20 

15 

4.3 

5.5 

6.6 

7.7 

8.9 

10.0 

11.1 

12.3 

134 

145 

15.7 

15 

20 

4.5 

5.6 

6.7 

7.8 

8.9 

10.0 

11.1 

12.2 

13.3 

14.4 

15.5 

10 

25 

4.8 

5.8 

§9 

7.9 

9.0 

10.0 

11.0 

12.1 

13.1 

14.2 

15.2 

5 

I       0 

5.1 

6.0 

7.0 

8.0 

9.0 

10.0 

11.0 

12.0 

13.0 

14.0 

14,9 

XI      0 

5 

5.4 

6.3 

7.2 

8.2 

9.1 

100 

10.9 

11.8 

12.8 

13.7 

14.6 

25 

10 

5.7 

6.6 

7.4 

8.3 

9.2 

10.0 

10.8 

11.7 

12.6 

13.4 

14.3 

20 

15 

6.1 

6.9 

7.7 

8.5 

9.2 

10.0 

10.8 

11.5 

12.3 

13.1 

13.9 

15 

20 

6.6;    7.2 

7.9 

8.6 

9.3 

10.0 

10.7 

11.4 

12.1 

128 

13.4 

10 

25 

7.0 

7.6 

8.2 

8.8 

9.4 

10.0 

10.6 

11.2 

11.8 

12.4 

13.0 

5 

II      0 

7.5 

8.0 

8.5 

9.0 

9.5 

10.0 

10.5 

11.0 

11.5 

12.0 

12.5 

X       0 

5 

7.9 

8.4 

8.8 

9.2 

9.6 

10.0 

10.4 

10.8 

11.2 

11.6 

12.1 

25 

10 

8.4 

8.7 

g.i 

9.4 

9.7 

10.0 

10.3 

10.6 

10.9 

11.3 

11.6 

20 

15 

8.9 

9.1 

9.4 

9.6 

9.8 

10.0 

10.2 

10.4 

10.6 

10.9 

11.1 

15 

20 

9.4 

9.5 

9.7 

9.8 

9.9 

10.0 

10.1 

10.2 

10.3 

10.5 

10.6 

10 

25 

9.9 

9.9 

9.9 

10.0 

10.0 

10.0 

10.0 

10.0 

10.1 

10.1 

10.1 

5 

III    0 

10.4 

10.3 

10.2 

10.1 

.10.1 

10.0 

9.9 

9.9 

9.8 

9.7 

9.6 

IX      0 

5 

10.8 

10.7 

10.5 

10.3 

10.2 

10.0 

9.8 

9.7 

9.5 

9.3 

9.2 

S5 

10 

11.3 

11.0 

10.8 

10.5 

10.3 

10.0 

9.7 

9.5 

9.2 

9.0 

8.7 

20 

15 

11.7 

11.4 

11.0 

10.7 

10.3 

10.0 

9.7 

9.3 

9.0 

8.6 

8,3 

15 

20 

12.1 

11.7 

11.3 

10.9 

10.4 

10.0 

9.6 

9.1 

8.7 

8.3 

7,9 

10 

25 

125 

12.0 

11.5 

11.0 

10.5 

10.0 

9.5 

9.0 

8.5 

8.0 

7,5 

5 

IV    0 

12.9 

12.3 

11.7 

11.2 

10.6 

10.0 

9.4 

8.8 

8.3 

7.7 

7.1 

VIII  0 

5 

13.3 

12.6 

11.9 

11.3 

10.6 

10.0 

9.4 

8.7 

8.1 

7.4 

6.7 

26 

10 

13.6 

12.9 

12.1 

11.4 

10.7 

10.0 

9.3 

8.6 

7.9 

7.1 

6.4 

20 

15 

13.9 

13.1 

12.3 

11.5 

10.8 

10.0 

92 

8.5 

7,7 

6.9 

6.1 

15 

20 

14.1 

133 

12.5 

11.6 

108 

10.0 

9.2 

8.4 

7.5 

6.7 

5.9 

10 

25 

14.4 

13.5 

12.6 

11.7 

10.9 

10.0 

9.1 

8.3 

7.4 

6.5 

&.6 

5 

V      0 

14.6 

13.7 

12.7 

11.8 

10.9 

10.0 

9.1 

8.2 

7.3 

6.3 

5.4 

VII    0 

5 

14.7 

13.8 

128 

11.9 

10.9 

10.0 

9.1 

8.1 

7.2 

6.2 

5.3 

25 

10 

14.9 

13.9 

12  9 

120 

11.0 

10.0 

9.0 

8.0 

7.1 

6.1 

5.1 

20 

15 

150 

140 

13.0 

12.0 

11.0 

10.0 

9.0 

8.0 

7.0 

6.0 

5.0 

15 

20 

15.1 

14.1 

13.0 

120 

11.0 

100 

9.0 

8.0 

7.0 

5.9 

4.9 

10 

25 

15.1 

14.1 

13.1 

12.0 

11.0 

10  0 

9.0 

80 

69 

5.9 

4.9 

5 

VI    0 

15.1 

14.1 

13.1 
20' 

12.1 
30" 

11.0 
40" 

10.0 
50" 

9.0 

8.0 
70" 

6.9 
80- 

5.9 
90" 

4.9 
100" 

VI      0 

0' 

10" 

60" 

90 


TABLE  LXXI. 
Moon's  Horary  Motion  in  Longitude. 
Argument.     Anomaly  corrected. 


Os 

diff. 

Is 

diff.  i    II*    diff 

III« 

diff 

\\S 

diff 

30.6 
29.2 

27.8 
26.4 
25.1 
23  8 

diff 

1.4 
1.4 
1.4 
1.3 
1.3 

o 
30 

29 
28 
27 
26 
25 

o 
0 

1 

2 
3 
4 
5 

441.5 
441.5 
441.3 
441.1 
440.8 
440.4 

0.0 

0.1 
0.2 
03 
0.4 

404.1 
401.6 
399.2 
396.6 
394.0 
391.3 

„    1 

2.5 
2.4 
2.6 
2.6 
2.7 

309.3  1 

305.6 

301.9 

298.1 

294.4 

290.6 

3.7 

3.7 
3.8 
3.7 
3.8 

19.5.3 
191.6 

187.9 
184.3 
180.6 
177.0 

3.7 
3.7 
3.6 
3.7 
3.6 

95.8 
93.0 
90.2 
87.6 
84.9 
82.3 

2.8 
2.8 
2.6 
2.7 
2.6 

0.5 

2.7  1 

3.8 

3.6 

2.6 

1.2 

6 

7 
8 

439.9 
439.4 
438.7 

0.5 
0.7 
0.7 
0.8 

38S.6 
385.8 
383.0 

1 
2  8  ; 

2.8 
2.9 
3.0 

286.8 
283.0 
279.2 

3.8 
3.8 
3.8 
3.9 

173.4 
169.8 
166.3 

3.6 
3.5 
3.5 
3.5 

79.7 
77.1 
74.6 

2.6 
2.5 
2.5 

2.4 

22.6 
21.4 
20.3 

1.2 
1.1 

24 
23 

22 

9 
10 

43S.0 
437.2 

380.1 
377.1 

275.4 
271.5 

162.8 
159.3 

72.1 
69.7 

19.2 
182 

1.1 
1.0 

21 
20 

0.9 

3.0 

3.8 

3.5 

2.4 

1.0 

11 
12 
13 

436.3 
435.3 
434.2 

1.0 
1.1 
1.1 
1.3 

374.1 
371.1 
368.0 

3.0 
3.1 
3.2 
3.2 

267.7 
263.8 
260.0 

3.9 
3.8 
3.8 
3.9 

155.8 
152.4 
148.9 

3.4 
3.5 
3.4 
3.3 

67.3 
65.0 
62.7 

23 
2.3 
2.3 

2.2 

17.2 
16.3 
15.4 

0.9 
0.9 
0.8 
0.8 

19 
18 
17 

14 

433.1 

364.8 

256.2 

145.5 

60.4 

14.6 

16 

15 

431.8 

361.6 

2.52.3 

142.2 

58.2 

138 

1ft 

1.3 

3.2 

3.8 

3.3 

2.1 

0.7 

16 
17 
18 
19 
20 

430.5 
429.1 
427.6 
426.1 
424.5 

1.4 
1.5 
1.5 
1.6 

358.4 
355.1 
351.8 
348.4 
345.0 

3.3 
3.3 

3.4 
3.4 

248.5 
244.6 
240.8 
236.9 
233.1 

3.9 
3.8 
3.9 
3.8 

138.9 
135.6 
132.3 
129.1 
125.9 

3.3 
3.3 
3.2 
3.2 

56.1 
53.9 
51.9 
49.8 
47.9 

2.2 
2.0 
2.1 
1.9 

13.1 

12.4 
11.8 
11.2 
10.7 

0.7 
0.6 
0.6 
0.5 

14 
13 
12 
11 
10 

1.7 

3.4 

3.8 

3.2 

2.0 

10.5 

21 
22 
23 

24 
2.5 

422.7 
421.0 
419.1 
417.2 
415.2 

1.7 
1.9 
1.9 
2.0 

2.1 

341.6 
338.1 
334.6 
331.1 
327.5 

3.5 
3.5 
3.5 
3.6 

3.5 

229.3 
225.4 
221.6 
217.8 
214.0 

3.9 

3.8 
3.8 
3.8 

3.7 

122.7 
119.6 
116.5 
•113.4 
;  110.4 
1 

3.1 
3.1 
3.1 
3.0 
3.0 

45.9 
44.0 
42.2 
40.4 
3S.7 

1.9 
1.8 
1.8 
1.7 

1.7 

10.2 
9.8 
9.4 
9.1 

8.8 

1 

0.4 
0.4 
0.3 
0.3 
0.2 

9 
8 
7 
6 
5 

26 

27 
28 
29 
30 

413.1 
410.9 
408.7 
406.4 
404.1 

2.2 
2.2 
2.3 
2.3 

324.0 
320.3 
316.7 
313.0 
309.3 

3.7 
3.6 
3.7 
3.7 

210.3 
206.5 
202.8 
199.0 
195.3 

3.8 
3.7 
3.8 
3.7 

107.4 

104.5 

101.6 

98.7 

95.8 

2.9 
2.9 
2.9 
2.9 

37.0 
35.3 
33.7 
32.1 
30.6 

1.7 
1.6 
1.6 
1.5 

8.6 
8.4 

8.3 
8.2 
8.2 

0.2 
0.1 
0.1 
0.0 

4 
3 

2 

1 
0 

XIs    i 

Xs 

VIIIs 

VIJs 

Vis 

TABLE  LXXII. 

Maori's  Horary  Motion  in  Longitude. 
Arguments.   Sum  of  preceding  Equations,  and  Arg.  of  Variation. 


91 


0 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

550 

600 

S       0 

0  0 

4.5 

5.5 

6.5 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.7 

14.7 

15.7  16.7 

1  .  ° 
XII  0 

5 

4.6 

5.6 

6.6 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.6 

14.6 

15.6|l6.6 

25 

10 

4.8 

5.8 

6.8 

7.7 

8.7 

9.6 

10.6 

11.5 

12.5 

13.4 

14.4 

15.3 

16.3 

20 

15  5.3 

6.1 

7.0 

7.9 

8.8 

9.7:10.5 

11.4 

12.3 

13.1 

14.0  14.9 

15.8 

15 

20  5.8 

6.6 

7.4 

8.2 

8.9 

9.7 

10.5 

11.2 

12.0|12.8 

13.5  14.3 

15.1 

10 

25 

6.6 

7.2 

7.8 

8.5 

9.1 

9.7 

10.4 

11.0 

11.7 

12.3 

12.9 

13.6 

14.2 

5 

I   0 

7.4 

7.8 

8.3 

8.8 

9.3 

9.8 

10.3 

10.8 

11.3 

11.8 

12.3 

12.7 

13.2 

XI  0 

5 

8.3 

8.6 

8.9 

9.2 

9.5 

9.9 

10.2 

10.5 

10.8 

11.2 

11.5 

11.8 

12.1 

25 

10 

92 

9.3 

9.5 

9.6 

9.8 

9.9 

10.1 

10.2 

10.4 

10.5 

10.7 

10.8 

11.0 

20 

15 

10.2 

10.1 

10.1 

10.1 

10.0 

10.0 

10.0 

10.0 

9.9 

9.9 

9.9 

9.8 

9.8 

15 

20 

U.l 

10.9 

10.7 

10.5 

10.3 

10.1 

9.9 

9.7 

9.5 

92 

9.0 

8.8 

8.6 

10 

25 

12.1 

11.7 

11.3 

10.9 

10.5 

10.2 

9.8 

9.4 

9.0 

8.6 

8.3 

7.9 

7.5 

5 

II  0 

12.9 

12.4 

11.8 

11.3 

10.8 

10.2 

9.7 

9.1 

8.6 

8.1 

7.5 

7.0 

6.4 

X  0 

5 

13.7 

13.0 

12.3 

11.6 

11.0 

10.3 

9.6 

8.9 

8.2 

7.5 

6.9 

6.2 

5.5 

25 

10 

14.3 

13.5 

12.7 

11.9 

11.1 

103 

9.5 

8.7 

7.9 1  7.1 

6.3 

5.5 

4.7 

20 

15 

14.9 

14.0 

13  1 

12.2 

11.3 

10.4 

9.5 

8.6 

7.7  6.8 

5.8 

4.9 

4.0 

15 

20 

15.3 

14.3 

13.3 

12.3 

11.4 

'0.4 

9.4 

8.4 

7.5  6.5 

5.5 

4.5 

3.6 

10 

25 

15.5 

14.5 

13.5 

12.4 

11.4 

10.4 

9.4 

8.4 

7.4  6.3 

5.3 

4.3 

33 

5 

III  0 

15.6 

14.5 

13.5 

12.5 

11.4 

10.4 

9.4 

8.4 

7.3 

6.3 

5.3 

4.2 

3.2 

IX  0 

5 

15.4 

14.4 

13.4 

12.4 

11.4 

10.4 

9.4 

8.4 

7.4 

6.4 

5.4 

4.4 

3.3 

25 

10 

15.2 

14.2 

13.3 

12.3 

11.3 

10.4 

9.4 

8  5 

7.5 

6.5 

5.6 

4.6 

3.6 

20 

15 

14.8 

13.9 

13.0 

12.1 

11.2 

10.4 

9.5 

8.6 

7.7 

6.8 

5.9 

5.1 

4.2 

15 

20 

14.2 

13.4 

12.6 

11.9 

11.1 

10.3 

9.5 

8.8 

8.0 

7.2 

6.4 

5.6 

4.9 

10 

25 

13.5 

12.9 

122 

11.6 

10.9 

10.3 

9.6 

9.0 

8.4 

7.6 

7.0 

6.3 

5.7 

5 

IV  0 

12.7 

12.2 

11.7 

11.2 

10.7 

10.2 

9.7 

9.2 

8.7 

8.2 

7.7 

7.2 

6.7 

VIII 0 

5 

11.9 

11.5 

11.2 

10.8 

10.5 

10.1 

9.8 

9.5 

9.1 

8.8 

8.4 

8.1 

7.7 

25 

10 

10.9 

10.7 

10.6 

10.4 

10.2 

10.1 

9.9 

9.7 

9.6 

9.4 

9.2 

9.1 

8.9 

20 

15 

9.9 

9.9 

10.0 

10.0 

10.0 

10.0 

10.0 

100 

10.0 

10.0 

10.1 

10.1 

10.1 

15 

20 

8.9 

9.1 

93 

9.5 

9.7 

9.9 

10.1 

10.3 

10.5 

10.7 

10.9 

11.1 

11.3 

10 

25 

8.0 

8.4 

8.7 

9.1 

9.5 

9.9 

10.2  10.6 

11.0 

11.3 

11.7 

12.1 

12.5 

5 

V  0 

7.1 

7.6 

8.2 

8.7 

9.2 

9.8 

10.3  10.9 

11.4 

11.9 

12.5 

13.0 

13.6 

VII  0 

5 

6.3 

7.0 

7.6 

8.3 

9.0 

9.7 

10.4  11.1 

11.8 

12.5 

13.2 

13.9 

14.6 

25 

10 

5.6 

6.4 

7.2 

8.0 

8.8 

9.7 

10.5  11.3 

12.1 

13.0 

13.8 

14.6 

15.4 

20 

15 

5.0 

5.9 

6.8 

7.8 

8.7 

9.6  lO.C  11.5 

12.4  13.3 

14.3 

15.2 

16.1 

15 

20 

4.6 

5  6 

6.6 

7.6 

8.6 

9.6  10.6  11.6 

12.6 

13.6 

14.6 

15.7 

16.7 

10 

25 

4.3 

5.4 

6.4 

7.5 

8.5 

9.6  10.6  11.7 

12  7 

13.8 

14.9 

15.9 

17.0 

5 

VI  0 

4.2 

5.3 

6.4 

7.4 

8.5 

9.6  10.6  11.7 

12.8 

13.9 

14.9 

16.0 

17.1 

VI  0 

0 

50 

100 

150 

200 

250  300  350 

400 

450 

500 

550 

600 

99  TABLE  LXXIII. 

Moon^s  Horary  Motion  in  Longitude. 
Argument.     Argument  of  tlie  Variation. 


1       0^ 

I« 

II« 

IJI« 

IVs 

Vs 

0 

0 

77.2 

57.8 

20.3 

2.4 

21.5 

59.7 

0 

30 

1 

77.2 

56.7 

19.2 

2.5 

22.7 

60.9 

29 

2 

77.1 

55.5 

18.1 

2.6 

23.8 

62.0 

28 

3 

77.0 

54.3 

17.0 

2.7 

25.0 

63.1 

27 

4 

76.8 

53.1 

16.0 

2.9 

26.2 

64.2 

26 

5 

76.6 

51.8 

15.0 

3.1 

27.5 

65.3 

25 

6 

7G.4 

50.5 

14.1 

3.3 

28.7 

66.3 

24 

7 

76.1 

49.3 

13.2 

37 

30.0 

67  3 

23 

8 

75.7 

48.0 

12.3 

4.0 

31.3 

68.3 

22 

9 

75.3 

46.7 

11.4 

4.4 

32.6 

69. 2 

21 

10 

74.9 

45.4 

10.6 

4.9 

33.9 

70.1 

20 

11 

74.4 

44.1 

9.8 

5.3 

35.2 

70.9 

19 

12 

73.9 

42.8 

9.0 

5.9 

36.5 

71.7 

18 

13 

73.3 

41.5 

8.3 

6.4 

37.8 

72.5 

17 

14 

72.7 

40.2 

7.6 

7.0 

39.2 

733 

16 

15 

•    72.0 

3S.9 

7.0 

7.7 

40.5 

74.0 

15 

16 

71.3 

37.5 

6.4 

8.3 

41.8 

74.7 

14 

17 

70.6 

36.2 

5.8 

9.1 

43.2 

75.3 

13 

18 

69.8 

34.9 

5.3 

9.8 

44.5 

75.8 

12 

19 

69.0 

33.6 

4.8 

10.6 

45.8 

76.4 

11 

20 

68.1 

32.3 

4.4 

11.5 

47.2 

76.9 

10 

21 

67.2 

31.1 

4.0 

12.3 

48.5 

77.3 

9 

22 

66.3 

29.8 

3.7 

13.2 

49.8 

77.7 

8 

23 

65.3 

28.6 

3.3 

14.2 

51.1 

78.1 

7 

24 

64.4 

27.3 

3.1 

15.1 

52.4 

78.4 

6 

25 

63.4 

26.1 

2.9 

16.1 

53.6 

78.6 

5 

26 

62.3 

24.9 

2.7 

17.1 

54.9 

78.9 

4 

27 

61.2 

23.7 

2.5 

18.2 

56.1 

79.0 

3 

28 

60.1 

22.5 

2.5 

19.3 

57.3 

79.2 

2 

29 

59.0 

21.4 

2.4 

20.4 

58.5 

79.2 

1 

30 

57.8 

20.3 

2.4 

21.5 

59.7 

79.2 

0 

XIs 

Xs 

IXs- 

VIIIs 

VIIs 

Vis 

TABLE  LXXIV.  93 

Moon^s  Horary  Motion  in  Longitude. 
Arguments.    Arg.  of  Reduction  and  Sum  of  preceding  Equations. 


0 

50  100  15( 

"   \   "  \   " 
200  250  300 

1  "  1  " 
350  400 

i 

4.50 

500  550 

600 

650 

s  0 
0  0 

3.3 

3.1  2.9 

2.7 

2.5 

2.3  2.1 

"  1  " 
1.9  1.7 

1.5 

1.3 

M 

0.9 

0.7 

,  0 
XII  0 

5 

3.3 

3.1  |2.9 

2.7 

25 

2.3 

2  1 

1.9  1.7 

1.5 

1.3 

1.1 

0.9 

0.7 

25 

10 

3.2 

30 

2.8 

2.6 

2.4 

2.3 

2.1 

1.9 

1.7 

1.5 

13 

1.1 

1.0 

0.8 

20 

15 

3.1 

29 

2.8 

2.6 

2.4 

2.2 

2.1 

1.9 

1.7 

1.5 

1.4 

1.2 

1.0 

0.9 

15 

20 

3.0 

28 

2.7 

2.5 

2.4 

2.2 

2.1 

1.9 

1.8 

1.6 

1.5 

1  3 

1.1 

1.0 

10 

25 

2.8 

2.7 

2.6 

2.4 

2.3 

2.2 

2.1 

1.9 

1.8 

1.7 

1.5 

1.4 

1.3 

1.2 

5 

I   0 

2.6 

2.5 

2.4 

2.3 

2.2 

2.1 

2.0 

1.9 

1.8 

1.7 

1.6 

1.5 

1.4 

1.3 

XI   0 

5 

2.4 

2.4 

2.3 

2.2 

22 

2.1 

2.0 

2.0 

1.9 

18 

1.8 

1.7 

1.6 

1.6 

25 

10 

2.2 

2.2 

2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

1.8 

1.8 

1.8 

20 

1.5 

2.0 

2.0 

2.0 

20 

2.0 

2.0 

20 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

15 

20 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.1 

2.1 

2.1 

2.2 

2.2 

2.2 

10 

25 

1.6 

1.6 

1.7 

1.8 

1.8 

1.9 

20 

2.0 

2.1 

22 

2.2 

2.3 

2.4 

2.4 

5 

II  0 

1.4  1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

X   0 

5 

1.2 

1.3 

1.4 

1.6 

1.7 

1.8 

1.9 

2.1 

2  2 

23 

2.5 

2.6 

2.7 

28 

25 

10 

1.0 

1.2 

1.3 

1.5 

1.6 

1.8 

1.9 

2.1 

2.2 

2.4 

2.5 

2.7 

2.9 

3.0 

20 

15 

0.9 

1.1 

1.2 

1.4 

i.n 

1.8 

19 

2.1 

2.3 

2,5 

2.6 

2.8 

3.0 

3.1 

15 

20 

0.8 

1.0 

1.2 

1.4 

1.6 

1  7 

1.9 

2.1 

2.3 

•J  5 

2.7 

2.9 

3.0 

3.2 

10 

25 

0.7 

0.9 

1.1 

1.3 

1.5 

1.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.1 

3.3 

5 

III  0 

0.7 

0.9 

1.1 

1.3 

1.5 

1.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.1 

3.3 

IX   0 

5 

0.7 

0.9 

l.l 

1.3 

1.5 

1.7 

1.9 

2.1 

2.3 

2.5  2  7 

2.9 

3.1 

3.3 

25 

10 

08 

1.0 

1.2 

1.4 

l.G 

1.7 

1.9 

2.1 

2.3 

2.5  2  7 

2.9 

3.0  3.2 

20 

15 

0.9 

1.1 

1.2 

1.4 

1.0 

1.8 

1.9 

2.1 

2.3 

25 

2.6 

2.8 

3  0  3.1 

15 

20 

1.0 

1.2 

1.3 

1.5 

l.R 

1.8 

1.9 

2.1 

2.2 

2.4 

2.5 

2.7 

2.9  3.0 

10 

25 

1.2 

1.3 

1.4 

l.G 

1.7 

1.8 

1.9 

2.1 

2.2 

2.3 

2.5 

2.6 

2.7  2.8 

5 

IV  0 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6  2.7 

VIII  0 

5 

1.6 

1.6 

1.7 

1.8 

1.8 

1.9 

2.0 

2.0 

2.1 

22 

2.2 

2.3 

2  4  2.4 

25 

10 

1.8 

1.8 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.1 

2.1 

2.1 

•1  0 

2  2  2.2 

20 

15 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0  2.0 

15 

20 

2.2 

2.2  2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

1.8 

1  8  1.8 

10 

25 

2.4 

2.4  2.3 

2.2 

2.2 

2.1 

2.0 

2.0 

1.9 

1.8 

1.8 

1.7 

1.6  1.6 

5 

V  0 

2.6 

2.5 

2.4 

2.3  2.2 

2.1 

2.0 

1.9 

1.8 

1.7 

1.6 

1.5 

1.4  1.3 

Vll  0 

5 

2.8 

2.7 

2.6 

2.4 

2.3 

2.2 

2.1 

1.9 

1.8 

1.7 

1.5 

1.4 

1.3  1.2 

25 

10 

3.0 

2.8 

2.7 

2.5 

2.4 

2.2 

2.1 

1.9 

1.8 

1.6 

1.5 

1.3 

1.1 

1.0 

20 

15 

3.1 

2.9 

2.8 

2.6 

2.4 

2.2 

2.1 

1.9 

1.7 

1.5 

1.4 

1.2 

1.0 

0.9 

15 

20 

3.2 

3.0 

2.8 

2.6 

2.4 

2.3 

2.1 

1.9 

1.7 

1.5 

1.3 

1.1 

1.0 

0.8 

10 

25 

3.3 

3.1 

2.9 

2.7 

2.5 

2.3 

2.1 

1.9 

1.7 

1.5 

1.3 

1.1 

0.9 

0.7 

5 

VI  0 

3.3 

3.1 

2.9 

2.7 

2.5 

2.3 

2.1 

1.9 

1.7 

1.5 

1.3 

1.1 

0.9 

0.7 

VI   0 

0 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

550 

600 

650 

94         TABLE  LXXV. 

Mooii's  Horary  Motion  in  Long. 
Arar.     Arjj.  of  Reduction. 


TABLE  LXXVL 

Moon's  Horarij  Motion  in  Long. 

(Equation  of  tlie  second  order.) 

Arguments.     Aro's  of  Table   LXX. 


Os  Vis 

Is  VI  s  TsVIIIs 

o 
0 

2.1 

6.0 

14.0 

o 
30 

1 

2.1 

6.3 

14.2 

29 

2 

2.1 

6.5 

14.4 

28 

3 

2.1 

6.8 

14.7 

27 

4 

2.2 

7.0 

14.9 

26 

5 

2.2 

7.3 

15^1 

25 

6 

2.2 

7.5 

15.3 

24 

7 

2.3 

7.8 

15.5 

23 

8 

2.4 

8.1 

15.7 

22 

9 

2.5 

8.4 

15.9 

21 

10 

25 

8.6 

16.1 

20 

11 

2.6 

8.9 

162 

19 

12 

2.7 

9.2 

16.4 

18 

13 

2.9 

9.4 

16.6 

17 

14 

3.0 

9.7 

16.7 

16 

15 

3  1 

10.0 

16.9 

15 

16 

3.3 

10.3 

17.0 

14 

17 

3.4 

10.6 

17.1 

13 

18 

3.6 

108 

17.3 

12 

19 

3.8 

11.1 

17.4 

11 

20 

3.9 

11.4 

17.5 

10 

21 

4.1 

11.6 

17.5 

9 

22 

4.3 

11.9 

17.6 

8 

23 

4.5 

12  2 

17.7 

7 

24 

4.7 

12.5 

17.8 

6 

25 

4.9 

12.7 

17.8 

5 

26 

5.1 

13.0 

17.8 

4 

27 

5.3 

132 

17.9 

3 

28 

5.6 

13.5 

17.9 

2 

29 

5.8 

137 

17.9 

1 

30 

6.0 

14.0 

17.9 

0 

, 

Xls  Vs  Xs  IVs 

iXs  Ills 

Arg. 

0 

50 

"1 
100 

s             0 

0  0 

1  0 
Ii          0 

III  0 

IV  0 

V  0 

VI  0 

VII  0 

VIII  0 
!X      0 

X  0 

XI  0 

XII  0 

0.05 

0.08 
0.10 
0.10 
0.09 
0.07 

0.05 
0.03 
0.01 
0.00 
0.00 
0.02 
0.05 

0.05 
0.05 
0.05 
005 
0.05 
0.05 

0.05 
0.05 
0.05 
0.05 
0.05 
0.05 
0.05 

0.05 
0.02 
0.00 
0.00 
0.01 
0.03 

0.05 
0.07 
0.09 
0.10 
0.10 
0.08 
0.05 

0 

50 

100 

Constant  to  be  iiddpd  27'24".0. 

TABLE  LXXVII. 
Maori's  Hnranj  Mown  in  Longitude. 
(Equations  of  the  second  order.) 
Arguments.     Arguments  of  Tables  LXXII  and  LXXTV, 


Variation. 

Reduction. 

0 

100 

200 

300 

400 

500 

600 

0 

0.03 
0.01 
0.01 
0.01 

0.03 
0.05 
0.05 
0.05 
0.03 

600 

0.03 
0.05 
0.06 
0.05 

0.03 
0.01 
0.00 
0.01 
0.03 

S               8         ° 

0.  VI.      0 

1.  VII.    0 

I.  VII.  15 

II.  VIII.  0 

III.  IX.      0 

IV.  X.       0 

IV.  X.      15 

V.  XI.      0 

VI.  XII.    0 

0.14 
0.22 
0.23 

0.22 

0.14 
0.06 
0.05 
0.06 
0.14 

0.14 
0.19 
0.20 
0.19 

0.14 
0.09 
0.08 
0.09 
0.14 

0.14 
0.16 
0.17 
0.16 

0.14 
0.12 
0.11 
0.12 
0.14 

0.14 
0.13 
0.13 
0.13 

0.14 
0.15 
0.15 
0.15 
0.14 

0.14 
0.10 
0.10 
0.10 

0.14 
0.18 
0.18 
0.18 
0.14 

0.14 
0.06 
0.05 
0.07 

0.14 
0.21 
0.23 
0.22 
0.14 

0.14 
0.02 
0.01 
0.03 

0.14 
0.26 
0.28 
0.26 
0.14 

TABLE  LXXVIIL 


95 


Maori's  Hoi-ary  Motion  in  Longitude. 

(Equations  of  the  second  order.) 

Arguments.     Args.  of  Eveclion,  Anomaly,  Variation,  Reduction. 


1  Evec. 

Anom 

1  Var. 

Red. 

Evec. 

Anom 

1  Var. 

lied. 

S 

0 

o 
0 

0,16 

1.05 

0.34 

0.08 

0.16 

1.05 

0.34 

0.08 

s       ° 
XII       0 

5 

0.15 

0.93 

0.28 

009 

0.18 

1.17 

0.40 

0.06 

25 

10 

0.13 

0.81 

0.22 

0.10 

0.19 

1.28 

0.46 

0.05 

20 

15 

0.12 

0.70 

0.17 

0.11 

0.21 

1.40 

0.51 

0.04 

15 

20 

0.10 

0.59 

0.12 

0.12 

0.22 

1.50 

0.,56 

0.03 

10 

25 

0.09 

0.49 

008 

0.13 

0.24 

1.60 

0.60 

0.02 

5 

I 

0 

0.08 

0.40 

0.05 

0.14 

0.25 

1.70 

0.63 

0.01 

XI        0 

5 

0.07 

0.31 

0.02 

0.15 

0.26 

1.78 

0.66 

0.01 

25 

10 

0.05 

0.24 

0.01 

0.15 

0.27 

1.86 

0.67 

0.00 

20 

15 

0.04 

0.17 

0.01 

0.15 

0.28 

1.92 

0.67 

0.00 

15 

20 

0.03 

0.12 

0.01 

0.15 

0.29 

1.98 

0.67 

0.00 

10 

25 

0.03 

0.07 

0.03 

0.15 

0.30 

2.02 

0.65 

0.01 

5 

II 

0 

0.02 

0.04 

0.06 

0.14 

0.31 

2.05 

0.62 

0.01 

X          0 

5 

0.01 

0.02 

0.09 

0.13 

0.32 

2.08 

0.59 

0.02 

25 

10 

0.01 

0.00 

0.13 

0.12 

0.32 

2.09 

0.54 

0.03 

20 

15 

0.00 

0.00 

0.18 

0.11 

0.32 

2.10 

0.50 

0.04 

15 

20 

0.00 

0.00 

0.24 

0.10 

0.33 

2.09 

0.44 

0.05 

10 

25 

0.00 

0.02 

0,29 

0.09 

0.33 

2.08 

0.39 

0.06 

5 

III 

0 

0.00 

0.04 

0.35 

0.08 

0.33 

2.06 

0.33 

0.08 

IX         0 

5 

0.00 

0.07 

0.40 

0.06 

0.33 

2.03 

0.27 

0.09 

25 

10 

0.01 

0.10 

0,46 

0.05 

0.32 

2.00 

0.22 

0.10 

20 

15 

0.01 

0.14 

0.51 

0.04 

0.32 

1.96 

0.17 

0.11 

15 

20 

001 

0.18 

0.56 

0.03 

0.31 

1.91 

0.12 

0.12 

10 

25 

0.02 

0.23 

0.60 

0.02 

0.31 

1.87 

0.08 

0,13 

5 

IV 

0 

0.03 

0.28 

0.63 

0.01 

0.30 

1.82 

0.05 

0.14 

VIII      0 

5 

0.03 

0.34 

0.66 

0.01 

0.29 

1.76 

0,02 

0.15 

25 

10 

0.04 

0.39 

0.67 

0.00 

0.28 

1.70 

0,01 

0.15 

20 

15 

0.05 

0.45 

0.68 

0.00 

0.27 

1.64 

0,00 

0.15 

15 

20 

0.06 

0.52 

0.07 

0.00 

0.26 

1.58 

0,00 

0.15 

10 

25 

0.08 

0.58 

0.66 

0.01 

0.25 

1.5? 

0.02 

0.15 

5 

V 

0 

0.09 

0.64 

0.64 

0.01 

0.24 

1.45 

0.04 

0.14 

VII       0 

5 

0.10 

0.71 

0.60 

0.02 

0.23 

1.39 

0.08 

0.13 

25 

10 

0.11 

0.78 

0.56 

0.03 

0.22 

1.32 

0.12 

0.12 

20 

15 

0.12 

0.84 

0.51 

0.04 

0.20 

1.25 

0.16 

0.11 

15 

20 

0.14 

0.91 

0.46 

0.05 

0.19 

1.18 

0.22 

0.10 

10 

25 

0.15 

0.98 

0.40 

0.06 

0.18 

1.12 

0.28 

0,09 

5 

VI 

0 

0.16 

1.05 

0.34 

0.08 

0.16 

1.05 

0.34 

0.08 

VI         0 

06 


TABLE  LXXIX. 

Mooji's  Horary  Motion  in  Latitude, 
Ari^timerit.      Artj.  I   of  Latitiidt^ 


r- 

0« 

U 

lis 

Ills 

IV* 

\s 

o 
0 

378.0 

35 1  3 

289.2 

200.0 

110.8 

45.7 

o 
30 

1 

378.0 

352.7 

286.5 

196.9 

108.1 

41.2 

29 

2 

377.9 

351.1 

283.8 

1938 

105.4 

42.7 

28 

3 

377.8 

349.4 

281.0 

190.7 

102.8 

41.3 

27 

4 

377.6 

347.7 

278.3 

187.5 

100.2 

39.9 

26 

5 

377.3 

346.0 

275.5 

184.4 

97.7 

38.6 

25 

6 

377.0 

344.2 

272.6 

181. 3 

95.1 

373 

24 

7 

376.7 

343.3 

269.8 

178.2 

92.6 

36.1 

23 

8 

376.3 

340.5 

2fiG,9 

175.1 

90.2 

34.9 

22 

9 

375.8 

338.5 

264.0 

172.1 

87.7 

33.8 

21 

10 

375.3 

336.6 

261.1 

169.0 

85.3 

32.7 

20 

11 

374.7 

334.5 

258.1 

165.9 

830 

31.6 

19 

12 

374.1 

332.5 

255.2 

162  9 

80.7 

30.7 

18 

13 

373.5 

330.4 

252.2 

159.8 

78.1 

29.7 

17 

14 

372.7 

328.3 

249.2 

156.8 

76.1 

28.9 

16 

15 

372.0 

326.1 

246.2 

153.8 

73.9 

28.0 

15 

16 

371.1 

323.9 

243.2 

1508 

71.7 

27.3 

14 

17 

370.3 

321.9 

2402 

147.8 

69.6 

26.5 

13 

18 

369.3 

319.3 

237. 1 

144.8 

G7.5 

25.9 

12 

19 

368.4 

317.0 

2:M.l 

141.9 

65.5 

25.3 

11 

20 

3673 

314.7 

231.0 

138.9 

63.4 

24.7 

10 

21 

366.2 

3123 

227.9 

1300 

61.5 

24.2 

9 

22 

365.1 

30;).8 

221.9 

133.1 

59.5 

23.7 

8 

23 

363.9 

3074 

221.8 

130.2 

57.7 

23.3 

7 

24 

362.7 

304.9 

218.7 

127.4 

55.8 

23.0 

6 

25 

361.4 

302.3 

215.0 

124.5 

540 

22.7 

5 

26 

360.1 

299,8 

2125 

121.7 

52.3 

22.4 

4 

27 

35S.7 

297.2 

209.3 

119. (» 

50  6 

22.2 

3 

28 

357.3 

294.6 

206.2 

116.2 

48.9 

22. 1 

2 

29 

355.8 

291.9 

203.1 

113  5 

47.3 

22.0 

1 

30 

354.3 

289.2 

200.0 

110.8 

45.7 

22.0 

0 

XIs 

Xs 

IXs 

VII  I-- 

VIl5 

Vl3 

TABLE  LXXX. 

Mooii's  Horary  Motion  in  Latitude. 
Arguments.  Args.  V,  VI,  VII,  VIII,  IX,  X,  XI,  and  XII,  of  Lalilude. 


Arg. 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

Arg. 

0 

0.00 

0.50 

0.34 

0.00 

0.50 

0.04 

0.12 

0.08 

1000 

50 

0.01 

0.49 

0.33 

0.00 

0.49 

0.04 

0.12 

0.07 

950 

100 

0.04 

0.45 

0.30 

0.02 

0.45 

0.04 

0.11 

0.05 

900 

150 

0.09 

0.40 

0.27 

0.04 

0.40 

0.03 

0.10 

0.03 

850 

200 

0.16 

0.33 

0.22 

0.06 

0.33 

0.03 

0.08 

0.01 

800 

250 

0.23 

0.25 

0.17 

0.09 

0.25 

0.02 

0.06 

0.00 

750 

300 

0.30  0.17 

0.12 

0.12 

0.17 

0.01 

0.04 

0.01 

700 

350 

0.37  0.10 

0.07 

0.14 

0.10 

0.01 

0.02  0.03 

650 

400 

0.42  0.05 

0.04 

0.16 

0.05 

0.00 

0.0i;0.05 

600 

450 

0.45  0.01 

0.01 

0.18 

0.01 

0.00 

0.00,0.07 

550 

500 

L_ 

0.46  0.00 

O.OOIO.I8 

0.00 

0.00 

o.oo!o.o8 

500 

TABLE  LXXXI.     Moon's  Horary  Motion  in  Latitude.         97 

Arguments.     Preceding  equation,   and  Sum  of  equations  of  Horary 

Motion  iu  Longitude,  except  the  last  two. 


Pr. 
eq. 

0  " 

50" 

100" 

150" 

200" 

250" 

300" 

3.50' 

400" 

450" 

.500  ' 

550' 

600" 

650" 

1".6 

1".4 

l."l 

0".9 

0".6 

0".4 

0".l 

0''.2 

0".4 

0".7 

0".9 

I".2 

1".4 

1".7 

Diff. 

20 

59.0 

54.5 

50.0 

45.4 

40.9 

.36.4 

31.8 

27.3 

22.8 

18.2 

13.7 

9.1 

4.6 

0.1 

4.5 

30 

57.4 

53.1 

48.9 

44.6 

40.3 

36.0 

31.7 

27.4 

23.2 

18.9 

14.6 

10.3 

6.0 

1.7 

4.3 

40 

55.8 

51.8 

47.7 

43.7 

39.7 

35.6 

31.6 

27.6 

23.6 

19.5 

15.5 

11.5 

7.4 

3.4 

4.0 

50 

54.2 

50.4 

46.6 

42.9 

39.1 

35.3 

31.5 

27.7 

24.0 

20.2 

16.4 

12.6 

8.8 

5.1 

3.8 

60 

52,6 

49.1 

45.5 

42.0 

38.5 

34.9 

31.4 

27.9 

24.4 

20.8 

17.3 

13.8 

10.2 

6.7 

3.5 

70 

51.0 

47.7 

44.4 

41.1 

37.9 

34.6 

31.3 

28.0 

34.8 

21.5 

18.2 

14.9 

11.7 

8.4 

3.3 

80 

49.3 

46.3 

43.3 

40.3 

37.3 

34.2 

31.2 

28.2 

25.2 

22.1 

19.1 

16.1 

13.1 

10.0 

3.0 

90 

47.7 

45.0 

42.2 

39.4 

36.7 

33.9 

31.1 

28.3 

25.6 

22.8 

20.0 

17.3 

14.5 

11.7 

2.8 

100 

46.1 

43.6 

41.1 

38.6 

36.0 

33.5 

31.0 

28.5 

26.0 

23.4 

20.9 

18.4 

15.9 

13.4 

2.5 

110 

44.5 

42.2 

40.0 

37.7 

35.4 

33.2 

30.9 

28.6 

26.4 

24.1 

21.8 

19.6 

17.3 

15.0 

2.3; 

120 

42.9 

40.9 

38.9 

36.9 

34.8 

32.8 

30.8 

28.8 

26.8 

21.8 

22.7 

20.7 

18.7 

16.7 

2.0 

130 

41.3 

.39.5 

37.8 

36.0 

34.2 

32.5 

30.7 

28.9 

27.2 

25.4 

23.7 

21.9 

20.1 

18.4 

1.8 

140 

39.7 

38.2 

36.7 

35.1 

33.6 

32.1 

30.6 

29.1 

27.6 

26.1 

24.6 

23.0 

21.5 

20.0 

1.5 

150 

38  1 

36.8 

35.5 

34.3 

33.0 

31.8 

30.5 

29.2 

28.0 

26.7 

25.5 

24.2 

23.0 

21.7 

1.3 

160 

36.5 

35.4 

34.4 

33.4 

32.4 

31.4 

30.4 

29.4 

28.4 

27.4 

26.4 

25.4 

24.4 

23.3 

1.0 

170 

.34.8 

34.1 

33.3 

32.6 

31.8 

31.1 

30.3 

29.5 

28.8 

28.0 

27.3 

26.5 

25.8 

25.0 

0.8 

180 

33.2 

32.7 

32.2 

31.7 

31.2 

30.7 

30.2 

29.7 

29.2 

28.7 

28.2 

27.7 

27.2 

26.7 

0.5 

190 

31.6 

31.4 

31.1 

30.9 

30.6 

30.4 

30.1 

29.8 

29.0 

29.3 

29.1 

28.8 

28.6 

28.3 

0.3 

200 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

30.0 

0.0 

210 

28.4 

28.6 

28.9 

29.1 

29.4 

29.6 

29.9 

30.2 

30.4 

30.7 

30.9 

31.2 

31.4 

31.7 

0.3 

220 

26.8  27.3 

27.8 

28.3 

28.8 

29.3 

29.8 

30.3 

30.8 

31.3 

31.8 

32.3 

32.8 

33.3 

0.5 

230 

25.2  25.9 

26.7 

27.4 

28.2 

28.9 

29.7 

30.5 

31.2 

32.0 

32.7 

33.5 

34.2 

35.0 

0.8 

240 

23.5  24.6 

25.6 

26.6 

27.6 

28.6 

29.6 

30.6 

31.6 

32.6 

33.6 

34.6 

35.6 

36.7 

1.0 

250 

21.923.2 

24.5 

25.7 

27.0 

28.2 

29.5 

30.8 

32.0 

33.3 

34.5 

35.8 

37.1 

38.3 

1.3 

260 

20.3  21.8 

23.3 

24.9 

26.4 

27.9 

29.4 

30.9 

32.4 

33.9 

35.4 

37.0 

38.5 

40.0 

1.5 

270 

18.7 

20. .5'  22.2i  24.0 

25.8 

27.5 

29.3 

31.1 

32.8 

34.6 

36.3 

38.1 

39.9 

41.6 

1.8 

280 

17  1 

19.ll  21.1   23.1 

25.2 

27.2 

29.2 

31.2 

33.2 

35.2 

37.3 

39.3 

41.3 

43.3 

2.0 

290 

15.5 

17.8  20.0  22.3 

24.6 

26.8 

29.1 

31.4 

33.6 

35.9 

38.2 

40.4 

42.7 

45.0 

2.3 

300 

13.9 

16.4   18.9  21.4 

24.0 

26.5 

29.0 

31.5 

34.0 

36.6 

39.1 

41.6 

44.1 

46.6 

2.5 

310 

12.3 

15.0    17.81  20.6 

23.3 

26.1 

28.9 

31.7 

34.4 

37.2 

40.0 

42.7 

45.5 

48.3 

2.8 

320 

10.7 

13.7 

16.7 

19.7 

22.7 

25.8 

28.8 

31.8 

34.8 

37.9 

40.9 

43.9 

46.9 

50.0 

3.0 

330 

9.0 

12.3 

15.6 

18.9 

22.1 

25.4 

28.7 

32.0 

35.2 

38.5 

41.8 

45.1 

48.3 

51.6 

33 

340 

7.4 

109 

14.5 

18.0 

21.5 

25.1 

28.6 

32.1 

35.6 

39.2 

42.7 

46.2 

49.8 

53.3 

35 

350 

5.8 

9.6 

13.4 

17.1 

20.9 

24.7 

28.5 

32.3  36.0 

39.8 

43.6 

47.4 

51.2 

54.9 

3.8 

360 

4.2 

8.2 

12.3 

16.3 

20.3 

24.4 

28.4 

32.4 

36.4 

40.5 

44.5 

48.5 

52.6 

56.6 

4.0 

370 

2.6 

6.9 

11.1 

15.4 

19.7 

24.0 

28.3  32.6 

36.8 

41.1 

45.4 

49.7 

54.0 

58.3 

4.3 

380 

1.0 
0" 

5.5 
50" 

10.0 
100" 

14.6 
150" 

19.1 
200" 

23.6 
250" 

28. 2|  32.7 

37.2 
400" 

41.8 

46.3 

50.9 
550" 

55.4 
600" 

59.9 
650" 

4.5 

300" 

350" 

450" 

500" 

TABLE  LXXXIL     Moon's  Horary  Motion  in  Latitude. 
Argument.     Arg.  IL  of  Latitude. 


0 
0 

0* 

Is 

Us 

Ills 

IV* 

V* 

0 
30 

9.3 

8.7 

7.1 

5.0 

2.9 

1.3 

3 

9.3 

8.6 

6.9 

4.8 

2.7 

1.2 

27 

6 

9.2 

8.5 

6.7 

4.6 

2.5 

1.1 

24 

9 

9.2 

8.3 

6.5 

4.3 

2.3 

1.0 

21 

12 

9.2 

8.2 

6.3 

4.1 

2.1 

0.9 

18 

15 

9.1 

8.0 

6.1 

3.9 

2.0 

0.9 

15 

18 

9.1 

7.9 

5.9 

3.7 

1.8 

0.8 

12 

21 

9.0 

7.7 

5.7 

3.5 

1.7 

0.8 

9 

24 

8.9 

7.5 

5.4 

3.3 

1.5 

0.8 

6 

27 

8.8 

7.3 

5.2 

3.1 

1.4 

0.7 

3 

30 

8.7 

7.1 

50 

2.9 

1.3 

0.7 

0 

Xs 

IXs 

VIII* 

VII5 

Vis 

M 


98  TABLE  LXXXIII. 

Moon's  Horary  Motion  in  Latitude. 

Arguments.  Preceding  equation,  and  Sum 
of  equations  of  Horary  Motion  in  Longi- 
tude, except  tiie  last  two. 


Prec. 

„ 

// 

,/ 

./ 

" 

'■ 

" 

a 

equ. 

0 

100 

200 

300 

400 

500 

600 

VOO 

0 

2.1 

1.8 

1.5 

1.2 

0.9 

0.6 

0.3 

0.0 

1 

1.9 

1.6 

1.4 

0.9 

0.7 

0.4 

0.2 

2 

1.7 

1.5 

1.3 

1.0 

0.8 

0.6 

0.3 

3 

1.5 

1.4 

1.2 

1.0 

0.9 

0.8 

0.6 

4 

1.3 

1.2 

1.2 

1.1 

1.0 

0.9 

0.9 

5 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

6 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

1.3 

7 

0.7 

0.8 

1.0 

1.2 

1.3 

1.4 

1.6 

8 

0.5 

0.7 

0.9 

1.2 

1.4 

1.6 

1.9 

9 

0.3 

0.6 

0.8 

1.3 

1.5 

1.8 

2.0 

10 

0.1 

0.4 

0.7 

1.0 

1.3 

1.6 

1.9 

2.2 

0 

100 

200 

300 

400 

500 

600 

700 

Constant  to  be  subtracted  237"  .2 

TABLE  LXXXV. 

Moon's  Horary  Motion  in  Latitude. 
(Equations  of  second  order.) 
Arguments.     Preceding  equation,  and  Sum 
of  equations  of  Horary  Motion  in  Longi- 
tude, except  the  last  two. 


Prec, 

" 

" 

" 

" 

" 

" 

" 

" 

equ. 

0 

100 

200 

300 

400 

500 

600 

700 

0.00 

0.65 

0.57 

0.48 

0.39 

0.31 

0.21 

0.12 

0.00 

0.10 ; 

0.62 

0.55 

0.47 

0.39 

0.31 

0.23 

0.15 

0.04 

0.20 

0.69 

0.53 

0.46 

0.39 

0.32 

0.25 

0.18 

0.09 

0.30  0.66] 

0.51 

0.45 

0.39 

0.33 

0.27 

0.21 

0.13 

0.40 

0.63 

0.48 

0.44 

0.39 

0.34 

0.29 

0.24 

0.17 

0.50 

0.50 

0.46 

0.43 

0.38 

0.35 

0.30 

0.27 

0.21 

0.60 

0.47 

0.44 

0.42 

0.38 

0.36 

0.32 

0.29 

0.25 

0.70  0.44 

0.42 

0.40 

0.38 

0.36 

0.34 

0.32 

0.30 

0.80  0.41 

0.40 

0.39 

0.38 

0.37 

0.36 

0.35 

0.34 

0.90 

0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

1.00 

0.35 

0.36 

0.37 

0.38 

0.39 

0.40 

0.41 

0.42 

1.10 

0.32 

0.34  ,0.36 

0.38 

0.40 

0.42 

0.44 

0.46 

1.20 

0.29 

0.32 

0.34 

0.38 

0.40 

0.44 

0.47 

0.51 

1.30 

0.26 

0.30 

033 

0.38 

0.41 

0.46 

0.49 

0.55 

1.40 

0.23 

0.28 

0.32 

0.37 

0.42 

0.47 

0.52 

0.59 

1.50 

0.20 

0.25 

0.31 

0.37 

0.43 

0.49 

0.55 

0.63 

1.60 

0.17 

0.2:5 

0.30 

0.37 

0.44 

0.51 

0.58 

0.67 

1.70 

0.14 

0.21 

0.29 

0.37 

0.45 

0.53 

0.61 

0.72 

1.80 

0.11 

0.19 

|0.28 

0.37 

0.45 

0.55 

0.64 

0.76 

0 

100 

200 

300 

400 

500 

600 

700 

TABLE  LXXXIV. 

Moon's  Hor.  Motion  in  Lot 

(Equa.  of  second  order.) 

Argument,     Arg.  I  of  Lat, 


I 

I 

e  o 

" 

" 

»  o 

0  0 

0.90 

0.90 

XII   0 

5 

0.83 

0.97 

25 

10 

0.75 

1.05 

20 

15 

0.68 

1.12 

15 

20 

0.61 

1.19 

10 

25 

0.54 

1.26 

5 

I   0 

0.47 

1.33 

XI   0 

5 

0.41 

1.39 

25 

10 

0.35 

1.45 

20 

15 

0.29 

1.51 

15 

20 

0.24 

1.56 

10 

25 

0.20 

1.60 

5 

II  0 

0.16 

1.64 

X    0 

5 

0.12 

1.68 

25 

10 

0.09 

1.71 

20 

15 

0.07 

1.73 

15 

20 

0.05 

1.75 

10 

25 

0.04 

1.76 

5 

III  0 

0.04 

1.76 

IX   0 

5 

0.04 

1.76 

25 

10 

0.05 

1.75 

20 

15 

0.07 

1.73 

15 

20 

0.09 

1.71 

10 

25 

0.12 

1.68 

5 

IV  0 

0.16 

1.64 

VIII  0 

5 

0.20 

1.60 

25 

10  0.24 

1.56 

20 

15  '0.29 

1.51 

15 

20  '0.35 

1.45 

10 

25  :0.41 

1.39 

5 

V  0  0.47 

1.33 

VII   0 

5  'o.54 

1.26 

25 

10  0.61 

1.19 

20 

15  0.68 

1.12 

15 

20  0.75 

1.05 

10 

25  0.83 

0.97 

5 

VI  0  ,0.90 

0.90 

VI   0 

TABLE  LXXXVI. 

Mean  New  Maoris  and  Arguments,  in  January. 


99 


Mean  New 

Years. 

Moon  in. 
January. 

I. 

II. 

III. 

IV. 

N. 

d.  h.    m. 

1821 

2  17  59 

0092 

7859 

80 

78 

823 

1822 

21  15  32 

0602 

7182 

78 

66 

930 

1823 

11  0  20 

0304 

5787 

61 

55 

953 

1824  B 

29  21  53 

0814 

5110 

59 

43 

060 

1825 

18  6  41 

0516 

3716 

42 

32 

083 

1826 

7  15  30 

0218 

2321 

25 

21 

105 

1827 

26  13  3 

0728 

1644 

24 

09 

213 

1828  B 

15  21  51 

0430 

0250 

07 

98 

235 

1829 

4  6  40 

0131 

8855 

90 

87 

257 

1830 

23  4  12 

0642 

8178 

88 

75 

365 

1831 

12  13  1 

0343 

6784 

71 

64 

387 

1832  B 

1  21  50 

0045 

5389 

54 

53 

409 

1833 

19  19  22 

0555 

4712 

53 

42 

517 

1834 

9  4  11 

0257 

3318 

36 

31 

539 

1835 

28  1  43 

0768 

2641 

34 

19 

647 

1836  B 

17  10  32 

0409 

1246 

17 

08 

669 

1837 

5  19  20 

0171 

9852 

00 

97 

692 

1838 

24  16  53 

0681 

9175 

99 

85 

799 

1839 

14  1  42 

0383 

7780 

82 

74 

822 

1840  B 

3  10  30 

0085 

6386 

65 

63 

844 

1841 

21  8  3 

0595 

5709 

63 

51 

951 

1842 

10  16  51 

0297 

4314 

46 

40 

974 

1843 

29  14  24 

0807 

3637 

44 

28 

081 

1844  B 

18  23  13 

0509 

2243 

28 

17 

104 

1845 

7  8  1 

0211 

0848 

11 

06 

126 

1846 

26  5  34 

0721 

0171 

09 

94 

234 

1847 

15  14  22 

0423 

8777 

92 

84 

256 

1848  B 

4  23  11 

0125 

7382 

75 

73 

278 

1849 

22  20  43 

0635 

6705 

73 

61 

386 

1850 

12  5  32 

0337 

5311 

56 

50 

408 

1851 

1  14  21 

0038 

3916 

40 

39 

431 

1852  B 

20  11  53 

0549 

3239 

38 

27 

538 

1853 

8  20  42 

0251 

1845 

21 

16 

560 

1854 

«7  18  14 

0761 

1168 

19 

04 

668 

1855 

17  3  3 

0463 

9773 

02 

93 

690 

1856  B 

6  11  51 

0164 

8379 

85 

82 

713 

1857 

24  9  24 

0675 

7702 

84 

70 

820 

1858 

13  18  13 

0376 

6307 

67 

59 

843 

1859 

3  3  1 

0078 

4913 

50 

48 

865 

1860  B 

22  0  34 

0588  1  4236 

48 

36 

972 

1 

100 


TABLE  LXXXVII. 


Mean  Lunations  and  Changes  of  the  Arguments, 


Num 

Lunations. 

I. 

II. 

III. 

IV. 

N. 

d. 

h     m 

I 

2 

14 

18  22 

404 

5359 

58 

50 

43 

1 

29 

12  44 

808 

717 

15 

99 

85 

2 

59 

1  28 

1617 

1434 

31 

98 

170 

3 

88 

14  12 

2425 

2151 

46 

97 

256 

4 

118 

2  56 

3234 

2869 

61 

96 

341 

5 

147 

15  40 

4042 

3586 

76 

95 

426 

6 

177 

4  24 

4851 

4303 

92 

95 

511 

7 

206 

17  8 

5659 

5020 

7 

94 

596 

8 

236 

5  52 

6468 

5737 

22 

93 

682 

9 

265 

18  36 

7276 

6454 

37 

92 

767 

10 

295 

7  20 

8085 

7171 

53 

91 

852 

11 

324 

20  5 

8893 

7889 

68 

90 

937 

12 

354 

8  49 

9702 

8606 

83 

89 

22 

13 

383 

21  33 

510 

9323 

98 

88 

108 

TABLE  LXXXVIIL 


Number  of  Days  from   the  commencement  of  the  year 
to  the  first  of  each  month- 


Months. 

Com. 

Bis. 

January 
February 
March  . 

Days. 
0 
31 
59 

Days. 
0 
31 
60 

April  . 

May 

June 

90 
120 
151 

91 
121 
152 

July   . 

181 

182 

August . 

September 

October 

212 
243 
273 

213 
244 
274 

November 

304 

305 

December 

334 

335 

TABLE  LXXXIX. 
Equations  for  New  and  Full  Moon. 


101 


Ktg. 

I 

II 

Arg. 

I 

II 

Arg 

III 

IV 

Arg 

h    m 

h     m 

h    m 

k     m 

m 

m 

0 

4  20 

10  10 

5000 

4  20 

10  10 

25 

3 

31 

25 

100 

4  36 

9  36 

5100 

4  5 

10  50 

26 

3 

31 

24 

200 

4  52 

9  2 

5200 

3  49 

11  30 

27 

3 

30 

23 

300 

5  8 

8  28 

5300 

3  34 

12  9 

28 

3 

30 

22 

400 

5  24 

7  55 

5400 

3  19 

12  48 

29 

3 

30 

21 

500 

5  40 

7  22 

5500 

3  4 

13  26 

30 

3 

30 

20 

600 

5  55 

6  49 

5600 

2  49 

14  3 

31 

3 

30 

19 

700 

6  10 

6  17 

5700 

2  35 

14  39 

32 

4 

30 

18 

800 

6  24 

5  46 

5800 

2  21 

15  13 

33 

4 

29 

17 

900 

6  38 

5  15 

5900 

2  8 

15  46 

34 

4 

29 

16 

1000 

6  51 

4  46 

6000 

1  55 

16  18 

35 

4 

29 

15 

1100 

7  4 

4  17 

6100 

1  42 

16  48 

36 

5 

28 

14 

1200 

7  15 

3  50 

6200 

1  31 

17  16 

|37 

5 

28 

13 

1300 

7  27 

3  24 

6300 

1  19 

17  42 

38 

5 

27 

12 

1400 

7  37 

2  59 

6400 

1  9 

18  6 

39 

5 

27 

11 

1500 

7   47 

2  35 

6500 

0  59 

18  28 

40 

6 

26 

10 

1600 

7  55 

2  14 

6600 

0  50 

18  48 

41 

6 

26 

9 

1700 

8  3 

1  53 

6700 

0  42 

19  6 

42 

7 

25 

8 

1800 

8  10 

1  35 

6800 

0  34 

19  21 

43 

7 

25 

7 

1900 

8  16 

1  18 

6900 

0  28 

19  33 

44 

7 

24 

6 

2000 

8  21 

1  3 

7000 

0  22 

19  44 

45 

8 

23 

5 

2100 

8  25 

0  51 

7100 

0  17 

19  52 

46 

8 

23 

4 

2200 

8  29 

0  40 

7200 

0  14 

19  57 

47 

9 

22 

3 

2300 

8  31 

0  32 

7300 

0  11 

20  0 

48 

9 

21 

2 

2400 

8  32 

0  25 

7400 

0  9 

20  1 

49 

10 

21 

1 

2500 

8  32 

0  21 

7500 

0  8 

19  59 

•50 

10 

20 

0 

2600 

8  31 

0  19 

7600 

0  8 

19  55 

51 

10 

19 

99 

2700 

8  29 

0  20 

7700 

0  9 

19  48 

52 

11 

19 

98 

2800 

8  26 

0  23 

7800 

0  11 

19  40 

53 

11 

18 

97 

2900 

8  23 

0  28 

7900 

0  15 

19  29 

54 

12 

17 

96 

3000 

8  18 

0  36 

8000 

0  19 

19  17 

55 

12 

17 

95 

3100 

8  12 

0  47 

8100 

0  24 

19  2 

,56 

13 

16 

94 

3200 

8  6 

0  59 

8200 

0  30 

18  45 

57 

13 

15 

93 

3300 

7  58 

1  14 

8300 

0  37 

18  27 

58 

13 

15 

92 

3400 

7  50 

1  32 

8400 

0  45 

18  6 

59 

14 

14 

91 

3500 

7  41 

1  52 

8500 

0  53 

17  45 

60 

14 

14 

90 

3600 

7  31 

2  14 

8600 

1  3 

17  21 

61 

15 

13 

89 

3700 

7  21 

2  38 

8700 

1  13 

16  56 

62 

15 

13 

88 

3800 

7  9 

3  4 

8800 

1  25 

16  30 

63 

15 

12 

87 

3900 

6  58 

3  32 

8900 

1  36 

16  3 

64 

15 

12 

86 

4000 

6  45 

4  2 

9000 

]  49 

15  34 

65 

16 

11 

85 

4100 

6  32 

4  34 

9100 

2  2 

15  5 

66 

16 

11 

84 

4200 

6  19 

5  7 

9200 

2  16 

14  34 

67 

16 

11 

83 

4300 

6  5 

5  41 

9300 

2  30 

14  3 

68 

16 

10 

82 

4400 

5  51 

6  17 

9400 

2  45 

13  31 

69 

17 

10 

81 

4500 

5  36 

6  54 

9500 

3  0 

12  58 

70 

17 

10 

80 

4600 

5  21 

7  32 

9600 

3  16 

12  25 

71 

17 

10 

79 

4700 

5  6 

8  11 

9700 

3  32 

11  52 

72 

17 

10 

78 

4800 

4  51 

8  50 

9800 

3  48 

11  18 

73 

17 

10 

77 

4900 

4  35 

9  30 

9900 

4  4 

10  44 

74 

17 

9 

76 

5000 

4  20 

10  10 

10000 

4  20 

10  10 

75 

17 

9 

75 

102 


TABLE  XC. 


Mean  Right  Ascensions  and  Declinations  of  50  principal  Fixed 
Stars,  for  the  beginning  of  1 840. 


Stars'  Name. 

Mag 

Rig 

ht  Ascen. 

AnnualVar. 

Declination. 

Ann.  Var. 

1  Algenib 

2  P  Andromedae 

3  Polans 

4  Achcrnar 

5  a  Arietis 

2.3 
2 

2.3 
1 
3 

A 
0 

1 

1 
1 
1 

m      8 
5     0.31 
0  46.7 
2  10.38 
31  44.88 
58     9.94 

+    3*0775 
3.309 

16.1962 
2.2351 
3.3457 

O        /            " 

14  17  38.82  N 
34  46  17.2   N 
88  27  21.96  N 
58     3     5.13  S 
22  42  11.81  N 

-f  20.051 
19.35 
19.339 

—  18.473 
•f  17.455 

6  a  Ceti 

7  a  Persci 

8  Aldeharan 

9  Capella 
10       Rigd 

2.3 
2.3 

1 
1 
1 

2 
3 
4 
5 
5 

bi  55.34 

12  55.97 

26  44.77 

4  52.67 

6  51.09 

+    3.1257 
4.2280 
3.4264 
4.4066 
2.8783 

3  27  30.09  N 

49   17     8.74N 

16  10  56.82  N 

45  49  42  81N 

8  23  29.29  S 

+  14.561 

13.371 

7.949 

4.793 

—   4.620 

11  /?Tauri 

12  y  Orionis 

13  aColumbae 

14  a  Orionis 

15  Canopus 

2 
2 
2 
1 

1 

5 
5 
5 
5 
6 

16   10.96 
16  33.1 
33  51.52 
46  30.71 
20  24.18 

+    3.7820 
3.210 
2  1688 
3.2430 
1.3278 

28  27  58.20  N 

6  11  55.3    N 
34     9  47.41  S 

7  22  17.14N 
52  36  38.42  S 

+    3.825 
+    3.82 
—   2.291 
+    1.191 
1.778 

1 6  Siriz^ 

17  Castor 

18  Procyon 

19  Pollux 

20  aHydrae 

1 

3 

1.2 

2 

2 

6 
7 

7 
7 
9 

38     5.76 
24  23.06 
30  55.53 
35  31.07 
19  43.57 

+    2.6458 
3.8572 
3.1448 
3.6840 
2.9500 

16  30     4.79  S 
32  13  58.89  N 

5  37  48.92  N 
28  24  25.57  N 

7  58     4.83  S 

+    4.449 

—    7,206 

8.720 

8.107 

+  15.341 

21  Reguhis 

22  a  Ursae  Majoris 

23  /?  Leonis 

24  jffVirginis 

25  y  Ursae  Majoris 

1 

1.2 
2.3 
3.4 

2 

9 
10 
11 
11 
11 

59  50.93 
53  47.98 
40  53.69 
42  21.4 
45  22.93 

+    3.2220 
3.8077 
3.0660 
3.124 
3.1914 

12  44  49.70  N 
62  36  48.93  N 
15  28     1.16N 
2  40     2  6    N 
54  35     4  67  N 

—  17.356 
19.221 
19.985 
19.98 
20.014 

26a2Crucis 

27  Spica 

28  e  Centauri 

29  a  Draconis 

30  Arcturus 

2 

1 
2 
3.4 

1 

12 
13 
13 
14 
14 

17  43.7 

16  46.36 

57  18.0 

0     2.8 

8  21.96 

+    3.258 
3.1502 
3.491 
1.625 
2.7335 

62  12  47.  9S 
10  19  24.39  S 
35  34  41.9    S 
65     8  32.1    N 
20     1     7.67  N 

+  19.99 
18.945 
17.499 

—  17.37 
18.956 

31  a  2  Centauri 

32  a  2  Librae 

33  /J  Ursae  Minoris 

34  y  2  Ursae  Minoris 

35  a  Coronae  Borealis 

1 
3 
3 

3.4 

2 

14  28  47.84 
14  42     2.44 

14  51   14.66 

15  21     1,3 
15  27  54.87 

+    4.0086 
3.3088 

—  0.2787 

—  0.179 
+    2.5277 

60  10     6.24  S 
15  22  18.25  S 
74  48  34.18N 
72  24  14.1    N 
27  15  27.71  N 

+  15  152 
15.256 

—  14.712 
12.81 
12.361 

36  a  Serpentis 

37  /?Scorpii 

38  Ant  ares 

39  aHerculis 

40  a  Ophiuchi 

2.3 

2 

1 
3.4 

2 

15 
15 
16 
17 

17 

36  23.43 
56     8.68 
19  36  49 
7  21.30 
27  30.56 

+    2.9386 
3.4729 
3.G625 
2.7317 
2.7724 

6  56     2.80  N 
19  21  38.82  S 
26     4  13.13S 
14  34  41.43  N 
12  40  58.65  N 

—  11.770 
+  10.330 

8.519 

—  4.576 

2.844 

41  6  Ursae  Minoris 

42  Vega 

43  Allair 

44  a  2  Capricomi 

45  a  Cygni 

3 
1 
1 
3 
1 

18 
18 
19 
20 
20 

23  56.48 
31   31.19 
42  58.61 
9  10.34 
35  58.80 

—  19.2072 

+    2.0116 

2.9255 

3.3323 

2.0416 

86  35  28.89  N 
38  38   16.85  N 
8  27     0.21  N 
13     2     5.57  S 
44  42  41.38  N 

+    2.161 

2.742 

8.701 

—  10.705 

+  12.614 

46  a  Aquarii 

47  Fomalhaut 

48  /SPegasi 

49  Markab 

50  a  Andromedae 

3 

1 

2 
2 

1 

21 
22 
22 
22 
24 

57  33.93 
48  47.67 
56     1.1 
56  47.75 
0     7.72 

+    3.0835 
3.3114 
2.878 
2.9771 
30704 

1     5  38.00  S 
30  28     4.91  S 

27  13     1.7    N 
14  20  46.92  N 

28  12  27.06  N 

—  17.256 
19.092 

+  19.255 
19295 
20.056 

TABLE  XCI. 


103 


Constants  for  the  Aher?'ation  and  Nutation  in  Right  Ascension 
and  Declination  of  the  Stars  in  the  preceding  Catalogue 


Aberration.          1 

Nutation. 

<P 

M 

i 

N 

<P' 

M' 

0' 

N' 

s      °      ' 

s    °  ' 

s    °  ' 

s      O       ' 

1 

8  28  47 

0.1087 

7  27  12 

0.9657 

6  8  24 

0.0300 

5  28  30 

0.8381 

2 

8  13  39 

0.1830 

6  19  12 

1.0740 

6  19  53 

0.0838 

5  10  8 

0.8496 

3 

8  13  51 

1.6526 

5  16  57 

1.3052 

8  16   7 

1.3427 

5  10  22 

0.8493 

4 

8  5  20  1 

0.3801; 

10  26  46 

1.2798 

4  10  12 

0.0775 

5  0  31 

0.8629 

5 

7  28  26 

0.1397 

7  0  2 

0.8972 

6  11   1 

0.0695 

4  22  53 

0.8765 

6 

7  14  11 

0.1149 

8  23  S 

0.8678 

6  1  26 

0.0322 

4  8  16 

0.9078 

7 

7  9  30 

0.3020 

5  3  5 

1.0630 

6  18  13 

0.1849 

4  3  47 

0.9179 

8 

6  21  43 

0.1447 

7  23  12 

0.5760 

6  3  27 

0.0726 

3  17  54 

0.9502 

9 

6  12  51 

0.2875 

3  25  37 

0.9112 

6  5  46 

0.18.30 

3  10  29 

0.9605 

10 

6  12  20 

0.1355 

9  3  42 

1.0300 

5  28  47 

1.9966 

3  10  4 

0.9608 

11 

6  10  13 

0.1873 

4  19  21 

0.3917 

6  2  52 

0.1008 

3  8  19 

0.9626 

12 

6  10  6 

0.1340 

8  26  4 

0.7851 

6  0  40 

0.0441 

3  8  14 

0.9626 

13 

6  6  5 

0.2145 

9  4  24 

1.2348 

5  26  18 

1.8750 

3  4  57 

0.9648 

14 

6  3  13 

0.1361 

8  28  23 

0.7521 

6  0  15 

0.0481 

3  2  37 

0.9657 

15 

5  25  22 

0.3491 

8  25  53 

1.2960 

6  8  46 

1.6679 

2  26  15 

0.9657 

16 

5  21  21 

0.1501 

8  25  51 

1.1152 

6  1  51 

1.9658 

2  22  58 

0.9636 

17 

5  10  40 

0.2010 

1  2  17 

0.6620 

5  24  2 

0.1257 

2  14  6 

0.9535 

18 

5  9  6 

0.1297 

9  6  54 

0.8071 

5  28  47 

0.0414 

2  12  47 

0.9513 

19 

5  8  2 

0.1829 

0  14  32 

0.6052 

5  24  2 

0.1114 

2  11  53 

0.9499 

20 

4  12  39 

0.1158 

8  17  31 

0.9967 

6  3  41 

0.0081 

1  18  37 

0.9007 

21 

4  2  22 

0.1162 

10  3  47 

0.8457 

5  23  47 

0.0480 

1  7  59 

0.8782 

22 

3  18  7 

0.4366 

0  3  28 

1.2394 

4  18  58 

0.2407 

0  21  57 

0.8520 

23 

3  5  21 

0.1117 

10  6  20 

0.9621 

5  20  56 

0.0344 

0  6  35 

0.8393 

24 

3  4  57 

0.0958 

9  6  51 

0.9075 

5  28  25 

0.0253 

0  6  5 

0.8390 

25 

3  4  8 

0.3229 

11  17  28 

1.2298 

4  21  46 

0.1465 

0  5  5 

0.8388 

26 

2  25  19 

0.4261 

6  8  5 

1.2585 

7  16  2 

0.2089 

11  24  14 

0.8390 

27 

2  9  22 

0.1066 

8  3  31 

0.8862 

6  5  51 

0.0154 

11  5  6 

0.8559 

28 

1  28  40 

0.1942! 

6  7  12 

1.0176 

6  17  31 

0.1062 

10  23  8 

0.8760 

29 

1  27  53 

0.48241 

10  23  28 

1.2995 

3  25  50 

0.1090 

10  22  16 

0.8777 

30 

1  25  46 

0.1336| 

9  28  18 

1.0974 

5  18  49 

1.9937 

10  20  1 

0.8822 

31 

1  20  32 

0.4123 

5  7  54 

1.1820 

6  29  6 

0.2460 

10  14  36 

0.8937 

32 

1  17  26 

0.1273 

7  18  24 

0.6923 

6  6  29 

0.0593 

10  11  28 

0.9006 

33 

1  14  42 

0.69611 

10  15  5 

1.3087 

2  26  45 

0.2235 

10  8  47 

0.9066 

34 

1  7  20 

0.6386 

10  7  33 

1.3087 

2  27  7 

0.0960 

10  1  45 

0.9225 

35 

1  5  45 

0.1704 

9  22  28 

1.1785 

5  17  18 

1.9510 

10  0  18 

0.9257 

36 

1  3  43 

0.1237 

9  8  22 

0.9994 

5  27  30 

0.00.58 

9  28  26 

0.9298 

37 

0  28  58 

0.1485 

7  4  4 

0.6237 

6  5  20 

0.0795 

9  24  12 

0.9386 

38 

0  23  24 

0.1728 

5  27  59 

0.5816 

6  5  49 

0.1029 

9  19  21 

0.9478 

39 

0  12  13 

0.1451 

9  5  25 

1.0962 

5  27  45 

1.9742 

9  9  58 

0.9610 

40 

0  7  34 

0.1427 

9  3  4 

1.0786 

5  28  48 

1.9803 

9  6  9 

0.9642 

41 

11  23  47 

i  1.3571 

8  22  49 

1.2821 

11  19  31 

0.8257 

8  24  57 

0.9650 

42 

11  22  50 

0.2393 

8  24  29 

1.2545 

6  5  31 

1.8436 

8  24  10 

0.9644 

43 

11  6  15 

0.1309 

8  22  59 

1.0237 

0  2  16 

1.9988 

8  10  21 

0.9472 

44 

11  0  2 

0.1341 

9  29  33 

0.6961 

5  26  12 

0.0609 

8  4  55 

0.9368 

45 

10  23  29 

0.2668 

8  0  39 

1.2634 

6  28  32 

1.9042 

7  29  0 

0.9242 

46 

10  2  57 

0.1057 

9  2  31 

0.8988 

5  29  26 

0.0264 

7  8  37 

0.8794 

47 

9  19  26 

0.1638 

11  7  34 

1.0271 

5  13  8 

0.0765 

6  23  30 

0.8540 

48 

9  17  29 

0.1491 

7  17  0 

1.1171 

6  17  2 

0.0162 

6  21  13 

0.8511 

49 

9  17  17 

0.1120 

8  2  5 

1.0138 

6  8  23 

0.0157 

6  20  58 

0.8508 

50 

9  0  6 

0.1495 

7  6  42 

1.0785 

6  17  20 

0.0444 

G  0  8 

0.8380 

104 


TABLE  XCII. 


Mean  Longitudes  and  Latitudes  of  some  of  the  principal  Fixed 
Stars  for  the  beginning  of  1840,  with  their  Annual  Variations. 


Stars'  Name. 

Mag 

L 

ong 

ritude. 

Annual 
Var. 

Latitude. 

Annual 
Var. 

s 

c 

/ 

// 

„ 

O       '            " 

,. 

a  Arietis 

3 

1 

5 

25 

27.6 

50.277 

9  57  40.9  N 

+  0.161 

Aldebaran 

2 

7 

33 

5.9 

50.210 

5  28  38.0  S 

—  0.335 

Capella 

2 

19 

37 

17.8 

50.302 

22  51  44.4  N 

—  0.052 

Polaris 

2.3 

2 

26 

19 

20.1 

47.959 

66     4  59.5  N 

+  0.552 

Sirius 

3 

11 

52 

32.9 

49.488 

39  34    4.3  S 

+  0.319 

Canopus 

3 

12 

44 

59.6 

49.366 

75  50  57.6  S 

+  0.459 

Pollux 

2 

3 

21 

0 

22.0 

49.502 

6  40  20.2  N 

-h  0,255 

Regulus 

4 

27 

36 

13.2 

49.946 

0  27  38.3  N 

+  0.220 

Spica 

6 

21 

36 

29.2 

50.085 

2     2  29.7  S 

+  0.171 

Arcturus 

6 

22 

0 

4.7 

50.711 

30  51    17.5  N 

+  0.214 

Antarcs 

1 

8 

7 

31 

45.2 

50.120 

4  32  51.6  S 

4  0.424 

Altair 

1.2 

9 

29 

31 

5.9 

50.795 

29   18  37.3  N 

+  0.080 

Fomalhaut 

11 

1 

36 

22.0 

50.595 

21     6  49.7  S 

+  0.213 

Achcrnar 

11 

13 

2 

5.3 

50.346 

17     6   17.3  S 

—  0.083 

a  Pegasi 

2 

11 

21 

15 

24.7 

50.112 

19  24  40.9  N 

+  0.098 

TABLE  added  to  TABLE  XC. 

Mean  Right  Ascensions  and  Declinations  of  Polaris  and  &  Ursae 
Minoris  for  1830,  1840,  1850,  and  1860. 


Stars. 

Years 

Right  Asc. 

Ann.  Var. 

Declination. 

Ann.  Var. 

Polaris 
i  Ursae  Minoris 

1830 
1840 
1850 
1860 

1830 
1840 
1850 
1860 

O          '            " 

0  59  30.76 

1  2  10.32 
1     5     0.29 
1     8     1.79 

18  27     5.13 
18  23  53.03 
18  20  40.21 
18  17  26.77 

+  15.478 
16.470 
17.567 

18.784 

—  19.167 
19.241 
19.305 
19.360 

O         /            " 

88  24     8.82 
88  27  22.43 
88  30  35.40 
88  33  47.64 

86  35     5.70 
86  35  27.93 
86  35  47.36 
86  36     3.97 

+  19.371 
19.309 
19.240 
19.163 

+    2.363 
2.085 
1.805 
1.523 

TABLE  XCIII. 
Second  Differences. 


105 


Hours  &  Minutes. 

r 

2' 

3' 

4' 

5' 

6' 

r 

8' 

9' 

10' 

11' 

h     m 

h 

m 

'> 

" 

,' 

/. 

., 

./ 

,f 

,' 

,/ 

,, 

„ 

0  0 

12 

0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0  10 

11 

50 

0,4 

0.8 

1.2 

1.6 

2,0 

2.4 

2.9 

3.3 

3.7 

4.1 

4.5 

0  20 

11 

40 

0.8 

1.6 

2.4 

3.2 

4,1 

4.9 

5.7 

6.5 

7.3 

81 

8.9 

0  30 

11 

30 

1.2 

2.4 

3.6 

4.8 

6,0 

7.2 

8.4 

9,6 

10,8 

12,0 

13.2 

0  40 

11 

20 

1.6 

3.1 

4.7 

6.3 

7,9 

9.4 

11.0 

12.6 

14.2 

15,7 

17.3 

0  50 

11 

10 

1.9 

3.9 

5.8 

7.8 

9,7 

11.6 

13.6 

15.5 

17.4 

19,4 

21.4 

1  0 

11 

0 

2.3 

4.6 

6.9 

9,2 

11.5 

13.8 

16,0 

18.3 

20.6 

22,9 

25.2 

1  10 

10 

50 

2.6 

6.3 

7.9 

10.5 

13.2 

15.8 

18,4 

21.1 

23.7 

26.3 

29.0 

I  20 

10 

40 

3.0 

5.9 

8.9 

11.9 

14.8 

17,8 

20.7 

23.7 

26.7 

29.6 

32.6 

1  30 

10 

30 

3.3 

6.6 

9.8 

131 

16.4 

19.7 

23.0 

26,3 

29.5 

32.8 

36.1 

1  40 

10 

20 

3.6 

7.2 

10.8 

14,4 

17.9 

21.5 

25.1 

28,7 

32.3 

35,9 

39.5 

I  50 

10 

10 

3.9 

7.8 

11.6 

15,5 

19.4 

33.3 

27.2 

31,0 

34.9 

38.8 

42.7 

2  0 

10 

0 

4.2 

8.3 

12.5 

16,7 

20.8 

25.0 

29.2 

33.3 

37.5 

41.7 

45.8 

2  10 

9 

50 

4.4 

8.9 

13.3 

17.8 

22.2 

26.6 

31.1 

35,5 

40.0 

44.4 

48.8 

2  20 

9 

40 

4.7 

9.4 

14.1 

18,8 

23.5 

28.2 

32.9 

37,6 

42.3 

47.0 

51.7 

2  30 

9 

30 

4.9 

9.9 

14,8 

19,8 

24.7 

29.7 

34.6 

39.6 

44.5 

49.5 

54.4 

2  40 

9 

20 

5.2 

10.4 

15.6 

20.7 

25.9 

31.1 

36.3 

41.5 

46.7 

51.9 

57.0 

2  50 

9 

10 

5.4 

10.8 

16.2 

21.6 

27.1 

32.5 

37.9 

43.3 

48.7 

54.1 

69.5 

3  0 

9 

0 

5.6 

11.3 

16.9 

22.5 

28.1 

33.8 

39.4 

45.0 

50.6 

56,3 

61.9 

3  10 

8 

50 

5.8 

11.7 

17.5 

23.3 

29.1 

35.0 

40.8  46.6 

52.4 

58,3 

64.1 

3  20 

8 

40 

6.0 

12.0 

18.1 

24.1 

30.1 

36.1 

42.1  48.1 

54.2 

60.2 

66.2 

3  30 

8 

30 

6.2 

12.4 

18,6 

24.8 

31.0 

37.2 

43.4  49,6 

55.8 

62,0 

68.2 

3  40 

8 

20 

6.4 

12.7 

19,1 

25.5 

31.8 

38.2 

44.6 

50.9 

57.3 

63,7 

70.0 

3  50 

8 

10 

6.5 

13.0 

19,6 

26.1 

32.6 

39.1 

45,7 

52.2 

58.7 

65.2 

71.7 

4  0 

8 

0 

6.7 

13.3 

20.0 

26.7 

33.3 

40.0 

46.7 

53,3 

60.0 

66.7 

73.3 

4  10 

7 

50 

6.8 

13.6 

20.4 

27.2 

34,0 

40.8 

47.6 

54.4 

61.2 

68.0 

74.8 

4  20 

7 

40 

6.9 

13.8 

20.8 

27.7 

34,6 

41.5 

48.4  J55.4 

62.3 

69.2 

76.1 

4  30 

7 

30 

7.0 

14.1 

21.1 

28.1 

35,2 

42.2 

49.2 

56.2 

63.3 

70.3 

77.3 

4  40 

7 

20 

7.1 

14.3 

21.4 

28.5 

35.6 

42.8 

49.9 

57.0 

64.2 

71.3 

78.4 

4  50 

7 

10 

7.2 

14.4 

21.6 

28.9 

36.1 

43.3 

50.5 

57.7 

64.9 

72,2 

79.4 

5  0 

7 

0 

7.3 

14.6 

21.9 

29.2 

36.5 

43.8 

51.0 

58,3 

65.6 

72.9 

80.2 

5  10 

6 

50 

7.4 

14.7 

22.1 

29.4 

36,8 

44.1 

51.5 

58.8 

66.2 

73.6 

80.9 

5  20 

6 

40 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.9 

59.3 

66.7 

74.1 

81.5 

5  30 

6 

30 

7.4 

14.9 

22.3 

29,8 

37,2 

44.7 

52.1 

59.6 

67.0 

74,5 

81.9 

5  40 

6 

20 

7.5 

15.0 

22,4 

29,9 

37,4 

44,9 

52.3 

59.8  67.3 

74,8 

82.2 

5  50 

6 

10 

7.5 

15.0 

22,5 

30,0 

37.5 

45,0 

52.5 

60.0  67.4 

74,9 

82.4 

6  0 

6 

0 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0  |67.5 

75.0 

82.5 

N 


106 


TABLE  XCIII. 


Second  Differences. 


Hours  &Min.  110"  20"  30"  40"  50"  1 


h    tn  \   h  771  " 

0  0112  0  0.0 

0  10  I  11  50  0.1 

0  20,  11  40  0.1 


0  30  11  30  0.2  0.4 
0  40  11  20  10.3  0.5 
0  50  11  10  i  0.3  0.6 


1  0 
1  10 
1  20 

1  30 
1  40 
1  50 


0 
10 
20 

30 
40 
50 

0 
10 
20 

30 
40 
50 

0 
10 
20 

4  30 
4  40 

4  50 

5  0 
5  10 
5  20 

5  30 
5  40 

5  50 

6  0 


11  0 
10  50 
10  40 

10  30 
10  20 
10  10 

10  0 

9  50 
9  40 

9  30 
9  20 
9  10 

9  0 

8  50 
8  40 

8  30 
8  20 
8  10 

8  0 
7  50 
7  40 

7  30 
7  20 
7  10 

7  0 
6  50 
6  40 

6  30 

6  20 

6  1(J 

6  0 


0.4 


0.8 


0.4  0.9 
0.5  1.0 


0.8  1.6 
0.9|  1.7 

0.9:  1.8 


0.0 
0.2 
0.4 

0.6 
0.8 
1.0 

1.1 
1.3 
1.5 

1.6 
1.8 
1.9 

2.1 

2.2 
2.3 

2.5 
2.6 


3.1  3.9 


3.3 
3.5 
2.713.6 


0.9  1.9  j  2.8 
1.0|  1.9  i  2.9 
1.0  2.0  [3.0 

1.0  2.1  I  3.1 

1.1  2.1  I  3.2 
1.112.2  3.3 


1.1  2.2 

1.1  2.3 

1.2  2.3 


2.3!  3.5 


3.8 
3.9 
4.0 

4.1 
4.2 
4.3 

4.4 
4.5 
4.6 

4.7 
4.8 
4.8 

4.9 
4.9 
4.9 

5.0 
5.0 
5.0 
5.0 


4.1 
4.3 
4.5 

4.7 
4.9 
5.0 

5.2 
5.3 
5.4 

5.6 

5.7 
5.8 

5.9 
5.9 
6.0 

6.1 
6.1 
6.1 

6.2 
6.2 
6.2 


0.0 

0.0 
0.0 

0.0 
0.0 
0.0 

00 
0.0 
0.0 

0.1 
0.1 
0.1 

0.1 
0.1 
0.1 

0.1 
0.1 
0.1 

0.1 
0.1 
0.1 

0.1 
0.1 
O.I 

0.1 
0.1 
0.1 

!0.l 
0.1 
0.1 

0.1 
0.1 
0.1 

0.1 
0.1 
0.1 


6.3I  0.1 


2" 

3" 

4" 

.n 

00 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.1 

0.1 

0.2 

0.2 

0.1 

0.1 

0.2 

0.2 

0.1 

0.1 

0.2 

0.2 

0.1 

0.2 

0.2 

0.3 

0.1 

0.2 

0.2 

0.3 

0.1 

0.2 

0.3 

0.3 

0.1 

0.2 

0.3 

0.3 

0.1 

0.2 

0.3 

0.4 

0.2 

0.2 

0.3 

0.4 

0.2 

0.2 

0.3 

0.4 

0.2 

0.3 

0.3 

0.4 

0.2 

0.3 

0.4 

0.5 

0.2 

03 

0.4 

0.5 

0.2 

0.3 

0.4 

0.5 

0.2 

0.3 

0.4 

0.5 

0.2 

0.3 

0.4 

0.5 

0.2 

0.3 

0.4 

0.5 

0.2 

0.3 

0.4 

0.5 

0.2 

0.3 

0.4 

0.6 

0.2 

0.3 

0.5 

0.6 

0.2 

0.3 

0.5 

0.6 

0.2 

0.4 

0.5 

0.6 

0.2 

0.4 

0.5 

0.6 

0.2 

0.4 

0.5 

0.6 

0.2 

0.4 

0.5 

0.6 

0.2 

0.4 

0.5 

0.6 

0.2 

0.4 

0.5 

0.6 

0.2 

0.4 

0.5 

0.6 

02 

0.4 

0.5 

0.6 

0.2 

0.4 

0.5 

06 

0.2 

0.4 

0.5 

0.6 

0.0 
0.0 
0.1 

0.1 
0.2 
0.2 


0.0  0.0 '0.0 

0.0  0.1  '0.1 

0.1  0.1  0.1 


0.2  0.2 
0.2  0.2 
0.3,0.3 


0.2  0.3  0.3  I  0.3 
0.3  I  0.3  0.4  0.4 
0.3  0.3  0.4  0.4 


0.3 
0.4 
0.4 

0.4 
0.4 
0.5 

0.5 
0.5 
0.5 

0  6'0.7 


0.6 
0.6 

0.6 
0.6 
0.7 

0.7 
0.7 
0.7 

0.7 
0.7 
0.7 

0.7 
0.7 
0.7 

0.7 
0.7 
0.7 
0.7 


0.4 
0.4 
0.5 

0.5 
0.5 
0.5 


0.7 
0.7 

0,7 
0,7 
0.8 

0.8 
0.8 
0.8 

0.8 
0.8 
0.8 

0.9 
0.9 
0.9 

0.9 
0.9 
0.9 
0.9 


0.4 
0.5 
0.5 

0.6 
0.6 
0.6 


0.6  0.7 
0.6  10.7 
0.6  0.7 

07 
0.8 
0.8 


0.8 
0.8 
0.9 

0.9 
0.9 
0.9 

0.9 
1.0 
1.0 

1.0 
1.0 
1.0 

1.0 
1.0 
1.0 
1.0 


0.5 
0.5 
0.6 

0.6 
0.7 
0.7 

0.7 
0.8 
0.8 

0.8 
0.9 
0.9 

0.9 
1.0 
1.0 

1.0 
1.0 
1.0 

1.1 
1.1 
1.1 

1.1 
1.1 
1.1 

1.1 
1.1 
1.1 
1.1 


TABLE  XCIV. 
Third  Differences. 


107 


Time  after 

Time  after 

noon  or 

10" 

20" 

30" 

40" 

50" 

1' 

2' 

3' 

4' 

5' 

noon  or 

midnight. 

midnight. 

+ 

,r 

/, 

-, 

/. 

» 

„ 

" 

// 

„ 

_ 

Oh.  Om. 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

12h.   Om. 

0     30 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.5 

0.7 

0.9 

11     30 

I       0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.6 

1.0 

1.3 

1.5 

11       0 

1     30 

0.1 

0.1 

0.2 

0,3 

0.3 

0.4 

0.8 

1.2 

1.6 

2.1 

10    30 

2      0 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.9 

1.4 

1.9 

2.3 

10      0 

2     30 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

1.0 

1.4 

1.9 

2.4 

9     30 

3       0 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.9 

1.4 

1.9 

2.3 

9      0 

3     30 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.9 

1.3 

1.7 

2.2 

8     30 

4      0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.7 

1.1 

1.5 

1.9 

8      0 

4    30 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.6 

0.9 

1.2 

1.5 

7    30 

6      0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

04 

0.6 

0.8 

1.0 

7      0 

5     30 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0,3 

0.4 

0.5 

6     30 

6       0 

0,0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

6      0 

+ 

— 

TABLE  XCV. 
Fourth  Differences. 


Time 

after 

1 

Time  after 

noon  or 

10" 

20" 

30" 

40" 

50" 

1' 

2' 

3' 

noon  or 

midnight. 

midnight. 

h. 

m. 

- 

" 

" 

/' 

'. 

„ 

/. 

- 

h.     m. 

0 

0 

0,0 

0.0 

0,0 

0.0 

0.0 

0,0 

0.0 

00 

12       0 

0 

30 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.6 

11     30 

1 

0 

0.1 

0.1 

0.2 

0.3 

0.3 

0,4 

0.8 

1.2 

11       0 

1 

30 

0.1 

0.2 

0.3 

0.4 

0.5 

0,6 

1.2 

1.7 

10    30 

2 

0 

0.1 

0.2 

04 

0.5 

0.6 

0.7 

1.5 

2.2 

10      0 

2 

30 

0.1 

0.3 

0.4 

0.6 

0.7 

'0.9 

1.8 

2.7 

9     30 

3 

0 

0.2 

0.3 

0.5 

0.7 

0.9 

|1.0 

2.1 

3.1 

9       0 

3 

30 

0.2 

0.4 

0.6 

0.8 

0.9 

1.1 

2.3 

3.4 

8     30 

4 

0 

0.2 

0.4 

0.6 

0.8 

1.0 

1.2 

2.5 

3.7 

8       0 

4 

30 

0.2 

0.4 

0.7 

0.9 

1.1 

1.3 

2.6 

3.9 

7    30 

5 

0 

0.2 

0.5 

0.7 

0.9 

1.1 

1.4 

2.7 

4.1 

7      0 

5 

30 

0.2 

0.5 

0.7 

0.9 

1.2 

1.4 

2.8 

4.2 

6     30 

6 

0 

0.2 

0.5 

0.7 

0.9 

1.2 

1.4 

2.8 

4.2 

6      0 

108 


TABLE  XCVI. 


Logistical  Logarithms. 


0 

0 

1 

60 

1.7782 

2     3  1 

4 
240 

5  1 

6 

7 

8 

9 

0 

120 
1.4771 

180 

300 

360 

420 

480 

540 

1.3010 

1.1761 

1.0792 

1.0000 

9331 

8751 

8239 

1 

3.5563 

1.7710 

1.4735 

1.2986 

1.1743 

1.0777 

9988 

9320 

8742 

8231 

2 

3.2553 

1,7639 

1.4699 

1.2962 

1.1725 

1.0763 

9976 

9310 

8733 

8223 

3 

3.0792 

1.7570 

1.4664 

1.2939 

1.1707 

1.0749 

9964 

9300 

8724 

8215 

4 

2.9542 

1.7501 

1.4629 

1.2915 

1.1689 

1.0734 

9952 

9289  8715  | 

8207 

5 

2.8573 

1.7434 

1.4594 

1.2891 

1.1671 

1.0720 

9940 

9279 

8706 

8199 

6 

2.7782 

1.7368 

1.4559 

1.2868 

1.1654 

1.0706 

9928 

9269 

8697 

8191 

7 

2.7112 

1.7302 

1.4525 

1.2845 

1.1636 

1.0692 

9916 

9259 

8688 

8183 

8 

2.6532 

1.7238 

1.4491 

1.2821 

1.1619 

1.0678 

9905 

9249 

8679 

8175 

9 

2.6021 

1.7175 

1.4457 

1.2798 

1.1601 

1.0663 

9893 

9238 

8670 

8167 

10 

2.5563 

1.7112 

1.4424 

1.2775 

1.1584 

1.0649 

9881 

9228 

8661 

8159 

11 

2.5149 

1.7050 

1.4390 

1.2753 

1.1566 

1.0635 

9869 

9218 

8652 

8152 

12 

2.4771 

1.6990 

1.4357 

1.2730 

1.1549 

1.0621 

9858 

9208 

8643 

8144 

13 

2.4424 

1.6930 

1.4325 

1.2707 

1.1532 

1.0608 

9846 

9198 

8635 

8136 

14 

2.4102 

1.6871 

1.4292 

1.2685 

1.1515 

1.0594 

9834 

9188 

8626 

8128 

15 

2.3802 

1.6812 

1.4260 

1.2663 

1.1498 

1.0580 

9823 

9178 

8617 

8120 

16 

2.3522 

1.6755 

1.4228 

1.2640 

1.1481 

1.0566 

9811 

9168 

8608 

8112 

17 

2.3259 

1.6698 

1.4196 

1.2618 

1.1464 

1.0552 

9800 

9158 

8599 

8104 

18 

2.3010 

1.6642 

1.4165 

1.2596 

1.1447 

1.0539 

9788 

9148 

8591 

8097 

19 

2.2775 

1.6587 

1.4133 

1.2574 

1.1430 

1.0525 

9777 

9138 

8582 

8089 

20 

2.2553 

1.6532 

1.4102 

1.2553 

1.1413 

1.0512 

9765 

9128 

8573 

8081 

21 

2.2341 

1.6478 

1.4071 

1.2531 

1.1397 

1.0498 

9754 

9119 

8565 

8073 

22 

22139 

1.6425 

1.4040 

1.2510 

1.1380 

1.0484 

9742 

9109 

8556 

8066 

23 

2.1946 

1.6372 

1.4010 

1.2488 

1.1363 

1.0471 

9731 

9099 

8547 

8058 

24 

2.1761 

1.6320 

1.3979 

1.2467 

1.1347 

1.0458 

9720 

9089 

8539 

8050 

25 

2.1584 

1.6269 

1.3949 

1.2445' 

1.1331 

1.0444 

9708 

9079 

8530 

8043 

26 

2.1413 

1.6218 

1.3919 

1.2424 

1.1314 

1.0431 

9697 

9070 

8522 

8035 

27 

2.1249 

1.6168 

1.3890 

1.2403 

1.1298 

1.0418 

9686 

9060 

8513 

8027 

28 

2.1091 

1.6118 

1.3860 

1.2382 

1.1282 

1.0404 

9675 

9050 

8504 

8020 

29 

2.0939 

1.6069 

1.3831 

1.2362 

1.1266 

1.0391 

9664 

9041 

8496 

8012 

30 

2.0792 

1.6021 

1.3802 

1.2341 

1.1249 

1.0378 

9652 

9031 

8487 

8004 

31 

2.0649 

1.5973 

1.3773 

1.2320 

1.1233 

1.0365 

9641 

9021 

8479 

7997 

32 

2.0512 

1.5925 

1.3745 

1.2300 

1.1217 

1.0352 

9630 

9012 

8470 

7989 

33 

2.0378 

1.5878 

1.3716 

1.2279 

1.1201 

1.0339 

9619 

9002 

8462 

7981 

34 

2.0248 

1.5832 

1.3688 

1.2259 

1.1186 

1.0326 

9608 

8992 

8453 

7974 

35 

2.0122 

1.5786 

1.3660 

1.2239 

1.1170 

1.0313 

9597 

8983 

8445 

7966 

36 

2.0000 

1.5740 

1.3632 

1.2218 

1.1154 

1.0300 

9586 

8973 

8437 

7959 

37 

1.9881 

1.5695 

1.3604 

1.2198 

1.1138 

1.0287 

9575 

8964 

8428 

7951 

38 

1.9765 

1.5651 

1.3576 

1.2178 

1.1123 

1.0274 

9564 

8954 

8420 

7944 

39 

1.9652 

1.5607 

1.3549 

1.2159 

1.1107 

1.0261 

9553 

8945 

8411 

7936 

40 

1.9542 

1.5563 

1.3522 

1.2139 

1.1091 

1.0248 

9542 

8935 

8403 

7929 

41 

1.9435 

i.5520 

1.3495 

1.2119 

1.1076 

1.0235 

9532 

8926 

8395 

7921 

42 

1.9331 

1.5477 

1.3468 

1.2099 

1.1061 

1.0223 

9521 

8917 

8386 

7914 

43 

1.9228 

1.5435 

1.3441 

1.2080 

1.1045 

1.0210 

9510 

8907 

8378 

7906 

44 

1.9128 

1.5393 

1.3415 

1.2061 

1.1030 

1.0197 

9499 

8898 

8370 

7899 

45 

1.9031 

1.5351 

1.3388 

1.2041 

1.1015 

1.0185 

9488 

8888 

8361 

7891 

46 

1.8935 

1.5310 

1.3362 

1.2022 

1.0999 

1.0172 

9478 

8879 

8353 

7884 

47 

1.8842 

1.5269 

1.3336 

1.2003 

1.0984 

1.0160 

9467 

8870 

8345 

7877 

48 

1.8751 

1.5229 

1.^310 

1.1984 

1.0969 

1.0147 

9456 

8861 

8337 

7869 

49 

1.8661 

1.5189 

1.3284 

1.1965 

1.0954 

1.0135 

9446 

8851 

8328 

7862 

50 

1.8573 

1.5149 

1.3259 

1.1946 

1.0939 

1.0122 

9435 

8842 

8320 

7855 

51 

1.8487 

1.5110 

1.3233 

1.1927 

1.0924 

1.0110 

9425 

8833 

8312 

7847 

52 

1.8403 

1.5071 

1.3208 

1.1908 

1.0909 

!  1.0098 

9414 

8824 

8304 

7840 

53 

1.8320 

1.5032 

1.3183 

1.1889 

1.0894 

!  1.0085 

9404 

8814 

8296 

7832 

54 

1.8239 

1.4994 

1.3158 

1.1871 

1  1.0880 

1.0073 

9393 

8805 

'8288 

7825 

55 

1.8159 

1.4956 

1.3133 

1.1852 

1.0865 

1.0061 

9383 

8796 

8279 

7818 

56 

1.8081 

1.4918 

1.3108 

1.1834 

1.0850 

1.0049 

9372 

8787 

8271 

7811 

57 

1.8004 

1.4881 

1  3083 

1.1816 

1.0835 

1.0036 

9362 

8778 

8263 

7803 

58 

1.7929 

1.4844 

1.3059 

1.1797 

1.0821 

1.0024 

9351 

8769 

8255 

7796 

59 

1.7855  1.4808 

1.3034 

1.1779 

1.0806 

1.0012 

9341 

8760 

8247 

1  7789 

60 

1.77S2  1.4771 

1.3010 

I  1761 

1.0792  1.0000 

9331 

8751 

8239  7782] 

TABLE  XCVI. 


Logistical  Logarithms. 


109 


0 

10 

]] 

12 

13 

U 

15 

16 

17 

18 

1080 
5229 

19 
1140 

20 

21 

600 

H60 
7368 

720 
6990 

780 

840 

900 
6021 

960 

1020 
5477 

1200 
4771 

1260 
4559 

7782 

6642 

6320 

5740 

4994 

1 

7774 

7361 

6984 

6637 

6315 

6016 

5736 

5473 

5225 

4990 

4768 

4556 

2 

7767 

7354 

6978 

6631 

6310 

6011 

5731 

5469 

5221 

4986 

4764 

4552 

3 

7760 

7348 

6972 

6625 

6305 

6006 

5727 

5464 

5217 

4983 

4760 

4549 

4 

7753 

7341 

6966 

6620 

6300 

6001 

5722 

5460 

5213 

4979 

4757 

4546 

5 

7745 

7335 

6960 

6614 

6294 

5997 

5718 

5456 

5209 

4975 

4753 

4542 

6 

7738 

7328 

6954 

6609 

6289 

5992 

5713 

5452 

5205 

4971 

4750 

4539 

7 

7731 

7322 

6948 

6603 

6284 

5987 

5709 

5447 

5201 

4967 

4746 

4535 

8 

7724 

7315 

6942 

6598 

6279 

5982 

5704 

5443 

5197 

4964 

4742 

4532 

9 

7717 

7309 

6936 

6592 

6274 

5977 

5700 

5439 

5193 

4960 

4739 

4528 

10 

7710 

7302 

6930 

6587  j 

6269 

5973 

5695 

5435 

5189 

4956 

4735 

4525 

11 

7703 

7296 

6924 

6581  1 

6264 

5968 

5691 

5430 

5185 

4952 

4732 

4522 

12 

7696 

7289 

6918 

6576 

6259 

5963 

5686 

5426 

5181 

4949 

4728 

451«1 

13 

7688 

7283 

6912  1 

6570  ' 

6254 

5958 

5682 

5422 

5177 

4945 

4724 

4515 

14 

7681 

7276 

6906  i  6565 

6248 

5954 

5677 

5418 

5173 

4941 

4721 

4511 

15 

7674 

7270 

6900 

6559 

6243 

5949 

5673 

5414 

5169 

4937 

4717 

4508 

16 

7667 

'.'264 

6894 

6554 

6238 

5944 

5669 

5409 

5165 

4933 

4714 

4505 

17 

7660 

7257 

6888 

6548 

6233 

5939 

5664 

5405 

5161 

4930 

4710 

4501 

18 

7653 

7251 

6882 

6543 

6228 

5935 

5660 

5401 

5157 

4926 

4707 

4498 

19 

7646 

7244 

6877 

6538 

6223 

5930 

5655 

5397 

5153 

4922 

4703 

4494 

20 

7639 

7238 

6871 

6532 

6218 

5925 

5651 

5393 

5149 

4918 

4699 

4491 

21 

7632 

7232 

6865 

6527 

6213 

5920 

5646 

5389 

5145 

4915 

4696 

4488 

22 

7625 

7225 

6859 

6521 

6208 

5916 

5642 

5384 

5141 

4911 

4692 

4484 

23 

7618 

7219 

6853 

6516 

6203 

5911 

5637 

5380 

5137 

4907 

4689 

4481 

24 

7611 

7212 

6847 

6510 

6198 

5906 

5633 

5376 

5133 

4903 

4685 

4477 

25 

7604 

7206 

6841 

6505 

6193 

5902 

5629 

5372 

5129 

4900 

4682 

4474 

26 

7597 

7200 

6836 

6500 

6188 

5897 

5624 

5368 

5125 

4896 

4678 

4471 

27 

7590 

7193 

6830 

6494 

6183 

5892 

5620 

5364 

5122 

4892 

4675 

4467 

28 

7583 

7187 

6824 

6489 

6178 

5888 

5615 

5359 

5118 

4889 

4671 

4464 

29 

7577 

7181 

6818 

6484 

6173 

5883 

5611 

5355 

5114 

4885 

4668 

4460 

30 

7570 

7175 

6812 

6478 

6168 

5878 

5607 

5351 

5110 

4881 

4664 

4457 

31 

7563 

7168 

6807 

6473 

6163 

5874 

5602 

5347 

5106 

4877 

4660 

4454 

32 

7556 

7162 

6801 

6467 

6158 

5869 

5598 

5343 

5102 

4874 

4657 

4450 

33 

7549 

7156 

6795 

6462 

6153 

5864 

5594 

5339 

5098 

4870 

4653 

4447 

34 

7542 

7149 

6789 

6457 

6148 

5860 

5589 

5335 

5094 

4866 

4650 

4444 

35 

7535 

7143 

6784 

6451 

6143 

5855 

5585 

5331 

5090 

4863 

4646 

4440 

36 

7528 

7137 

6778 

6446 

6138 

5850 

5580 

5326 

5086 

4859 

4643 

4437 

37 

7522 

7131 

6772 

6441 

6133 

5846 

5576 

5322 

5082 

4855 

4639 

4434 

38 

7515 

7124 

6766 

6435 

6128 

5841 

5572 

5318 

5079 

4852 

4636 

4430 

39 

7508 

7118 

6761 

6430 

6123 

5836 

5567 

5314 

6075 

4848 

4632 

4427 

40 

7501 

7112 

6755 

6425 

6118 

5832 

5563 

5310 

5071 

4844 

4629 

4424 

41 

7494 

7106 

6749 

6430 

'6113 

5827 

5559 

5306 

5067 

4841 

4625 

4420 

42 

7488 

7100 

6743 

6414 

i6108 

5823 

5554 

5302 

5063 

4837 

4622 

4417 

43 

7481 

7093 

6738 

6409 

6103 

5818 

5550 

5298 

5059 

4833 

4618 

4414 

44 

7474 

7087 

6732 

6404 

16099 

5813 

5546 

5294 

5055 

4830 

4615 

4410 

Vi 

7467 

7081 

6726 

6398 

6094 

5809 

5541 

5290 

5051 

4826 

4611 

4407 

46 

7461 

7075 

6721 

6393 

6089 

5804 

5537 

5285 

5048 

4822 

4608 

4404 

47 

7454 

7069 

6715 

6388 

6084 

5800 

5533 

5281 

5044 

4819 

14604 

4400 

48 

7447 

7063 

6709 

6383 

6079 

5795 

5528 

5277 

5040 

4815 

4601 

4397 

49 

7441 

7057 

6704 

6377 

6074 

5790 

5524 

5273 

5036 

14811 

4597 

4394 

50 

7434 

7050 

6698 

6372 

6069 

5786 

5520 

5269 

5032 

4808 

4594 

4390 

51 

7427 

7044 

6692 

6367 

6064 

5781 

5516 

5265 

5028 

4804 

4590 

4387 

52 

7421 

7038 

6687 

6362 

6059 

5777 

5511 

5261 

5025 

1  4800 

4587 

4384 

53 

7414 

7033 

6681 

6357 

6055 

5772 

5507 

1  5257 

5021 

4797 

4584 

4380 

54 

7407 

7026 

6676 

0351 

i  6050 

5768 

5503 

5253 

,5017 

4793 

4580 

4377 

55 

7401 

7020 

6670 

6346 

6045 

5763 

5498 

1  5249 

5013 

4789 ' 4577 

4374 

56 

7394 

7014 

6664 

6341 

6040 

5758 

5494 

5245 

5009 

4786 

4573 

4370 

57 

7387 

7008 

6659 

6336 

6035 ' 5754 

5490 

5241 

5005 

4782 

1  4570 

4367 

58 

7381 

7002 

6653 

6331 

6030 

,  5749 

5486 

5237 

5002 

4778 

[4566 

4364 

59 

7374 

6996 

6648 

6325 

6025 

5745 

5481 

5233 

4998 

4775 

,4563 

4361 

60 

7368 

6990 

6642 

6320 

6021 

5740 

5477 

5229 

4994 

4771  4559 ' 4357 

_ 

no 


TABLE  XCVI.     Logistical  Logarithms. 


'  1  22  1  23  1  24 

25 

26 

27 

28 

29 

30 

31  1  32 

33 

■' 

1320  1380 

1440 

1500 
3802 

1560 
3632 

1620 

1680 

1740 

1800 

3oTo 

1860 

1920 

iJfcO 

0 

4357 

4164 

3979 

3463 

3310 

3158 

2868 

2730 

2596 

1  4354 

4161 

3976 

3799 

3G29 

3465 

3307 

3)55 

3008 

2866 

2728 

2594 

2  4331 

4158 

3973 

3796 

3620 

3463 

3305 

3153 

3005 

2863 

2725 

2592 

3  4347 

4155 

3970 

3793 

3623 

3460 

3302 

3150 

3003 

2861 

2723 

2590 

4  4344 

4152 

3967 

3791 

3621 

3457 

3300 

3148 

3001 

2859 

2721 

2583 

ft 

4341 

4149 

3964 

3788 

3618 

3454 

3297 

3145 

2998 

2856 

2719 

2585 

6 

4338 

4145 

3961 

3785 

3615 

3452 

3294 

3143 

2996 

2854 

2716 

2583 

7 

4334 

4142 

3958 

3782 

3612 

3449 

3292 

3140 

2993 

2852 

2714 

2581 

8  I  4331 

4139 

3955 

3779 

3610 

3446 

3289 

3138 

2991 

2849 

2712 

2579 

9  j  4328 

4136 

3952 

3776 

3607 

3444 

3287 

3135 

2989 

2847 

2710 

2577 

10  4335 

4133 

3949 

3773 

3604 

3441 

3284 

3133 

2986 

2845 

2707 

2574 

11  4321 

4130 

3946 

3770 

3601 

3438 

3282 

3130 

2984 

2842 

2705 

2572 

12  4318 

4127 

3943 

3768 

3598 

3436 

3279 

3128 

2981 

2840 

2703 

2570 

13  4315 

4124 

3940 

3765 

3596 

3433 

3276 

3125 

2979 

2838 

2701 

2568 

14  4311 

4120 

3937 

3762 

3593 

3431 

3274 

3123 

2977 

2835 

2698 

2566 

15  4308 

4117 

3934 

3759 

3590 

3428 

3271 

3120 

2974 

2833 

2696 

2564 

16  4305 

4114 

3931 

3756 

3587 

3425 

3269 

3118 

2972 

2831 

2694 

2561 

17  1  4302 

4111 

3928 

3753 

3585 

3423 

3266 

3115 

2969 

2828 

2692 

2559 

18 1 4298 

4108 

3925 

3750 

3582 

3420 

3264 

3113 

2967 

2826 

2689 

3557 

19 

4295 

4105 

3922 

3747 

3579 

3417 

3261 

3110 

2965 

2824 

2687 

2555 

20 

4292 

4102 

3919 

3745 

3576 

3415 

3259 

3108 

2962 

2821 

2685 

2553 

21 

4289 

4099 

3917 

3742 

3574 

3412 

3256 

3105 

2960 

2819 

2683 

2551 

22 

4285 

4096 

3914 

3739 

3571 

3409 

3253 

3103 

2958 

2817 

2681 

2548 

23 

4282 

4092 

3911 

3736 

3568 

3407 

3251 

3101 

2955 

2815 

2678 

2546 

24 

4279 

4089 

3908 

3733 

3565 

3404 

3248 

3098 

2953 

2812 

2676 

2544 

25 

4276 

4086 

3905 

3730 

3563 

3401 

3246 

3096 

2950 

2810 

2674 

2542 

26 

4273 

4083 

3902 

3727 

3560 

3399 

3243 

3093 

2948 

2808 

2672 

2540 

27 

4269 

4080 

3899 

3725 

3557 

3396 

3241 

3091 

2946 

2805 

2669 

2538 

28 

4266 

4077 

3896 

3722 

3555 

3393 

3238 

3088 

2943 

2803 

2667 

2535 

29 

4263 

4074 

3893 

3719 

3552 

3391 

3236 

3086 

2941 

2801 

2665 

2533 

30 

4260 

4071 

3890 

3716 

3549 

3388 

3233 

3033 

2939 

2798 

2663 

2531 

31 [4256 

4068 

3887 

3713 

3546 

33S6 

3231 

3081 

2936 

2796 

2660 

2529 

32 

4253 

4065 

3884 

3710 

3544 

3383 

3228 

3073 

2934 

2794 

2658 

2527 

33 

4250 

4062 

3881 

3708 

3541 

3330 

3235 

3076 

2931 

2793 

2656 

3525 

34 

4247 

4059 

3878 

3705 

3533 

3378 

3223 

3073 

2929 

2739 

2654 

a522 

35 

4244 

4055 

3375 

3702 

3535 

3375 

3220 

3071 

3927 

2787 

2652 

2520 

36 

4240 

4052 

3872 

3699 

3533 

3372 

3218 

3069 

2924 

2735 

2649 

2518 

37 

4237 

4049 

3869 

3696 

3530 

3370 

3315 

3066 

2922 

2782 

2647 

2516 

33 

4234 

4046 

3366 

3693 

3527 

3367 

3213 

3064 

2920 

2780 

2645 

2514 

39 

4231 

4043 

3863 

3691 

3525 

3365 

3210 

3061 

2917 

2778 

2643 

2512 

40 

4228 

4040 

3360 

3688 

3522 

3362 

3203 

3059 

2915 

2775 

2640 

2510 

41 

4224 

4037 

3857 

3685 

3519 

3359 

3305 

3056 

2912 

2773 

2638 

2507 

42 

4221 

4034 

3855 

3682 

3516 

3357 

3303 

3054 

2910 

3771 

2636 

2305 

43 

4218 

4031 

3852 

3679 

3514 

3354 

3300 

3052 

2903 

2769 

2634 

3503 

44 

4215 

4028 

3849 

3677 

3511 

3351 

3198 

3049 

2905 

2766 

2632 

2301 

45  4212 

4025 

3340 

3674 

3503 

3319 

3195 

3047 

2903 

2764 

2629 

2499 

46 

4209 

4022 

3343 

3671 

3506 

3316 

3193 

3044 

2901 

2762 

2527 

2497 

47 

4205 

4019 

3340 

3668 

3503 

3344 

3190 

3042 

2898 

2760 

2625 

2494 

48 

4202 

4016 

3337 

3665 

3500 

3341 

3188 

3039 

2896 

2757 

2623 

2492 

49 

4199 

4013 

3834 

3603 

3497 

3338 

3185 

3037 

2894 

2755 

2621 

2490 

50 

4196 

4010 

3831 

3660 

3495 

3336 

3183 

3034 

2891 

2753 

2618 

2488 

51 

4193 

4007 

3828 

3657 

3492 

3333 

3180 

3032 

2889 

2750 

2616 

2486 

52 '4189 

4004 

3825 

3654 

3489 

3331 

3178 

3030 

2887 

2748 

2614 

2484 

53  I  4186 

4001 

3322 

3651 

3487 

3328 

3175 

3027 

2884 

2746 

2612 

2482 

54 

4183 

3998 

3820 

3649 

3484 

3325 

3173 

3025 

2882 

2744 

2610 

2480 

55 

4180 

3995 

3317 

3646 

3481 

3323 

3170 

3022 

2880 

2741 

2607 

2477 

56 

4177 

3991 

3814 

3643 

3479 

3320 

3168 

3020 

2877 

2739 

2605 

2475 

57 

4174 

3988 

3311 

3640 

3476 

3318 

3165 

3018 

2875 

2737 

2603 

2473 

58 

4171 

3985 

3808 

3637 

3473 

3315 

3163 

3015 

2873 

2735 

2601 

2471 

59 

4167 

3982 

3805 

3635 

3471 

3313 

3160 

3013 

2870 

2732 

2599 

2469 

60  14164 

3979 

3802 

3632  3468 

3310 

3158 

3010 

2308 

2?30 

2596 

2467 

TABLE  XCVI.     Logistical  Logarithms. 


Ill 


u 

34 
2040 

35 
2l0b 

36  1  37 

38 

39 

40 
2400 
1761 

41 

2460 

42 
"2520 

43  1  44 

45 

2160  2»20 

2280 
l'J84 

2340 
1871 

2£80j  2fi40 

2700 
1249 

2467 

2341 

2218 

2099 

1654 

1549 

1447  1347 

1 

2465 

2339 

2210 

2098 

1982 

1869 

1759 

1652 

1547 

1445  1345 

1248 

2 

2462 

2337 

2214 

2096 

1980 

1867 

1757 

1650 

1546 

1443  1344 

1246 

3 

2460 

2335 

2212 

2094 

1978 

1865 

1755 

1648 

1544 

1442  1342 

1245 

4 

2458 

2333 

2210 

2092 

1976 

1863 

1754 

1647 

1542 

1440 

1340 

1243 

5 

2456 

2331 

£208 

2090 

1974 

1862 

1752 

1645 

1540 

1438 

1339 

1241 

6 

2454 

2328 

2206 

2088 

1972 

1860 

1750 

1643 

1539 

1437 

1337 

1240 

7 

2452 

2326 

2204 

2086 

1970 

1858 

1748 

1041 

1537 

1435 

1335 

1238 

8 

2450 

2324 

2202 

2084 

1968 

1856 

1746 

1640 

1535 

1433 

1334 

1237 

9 

2448 

2322 

2200 

2082 

1967 

1854 

1745 

1638 

1534 

1432 

1332 

1235 

10 

2445 

2320 

2198 

2080 

1965 

1852 

1743 

1636 

1532 

1430 j  1331 

1233 

11 

2443 

2318 

2196 

2078 

1963 

1850 

1741 

1634 

1530 

1428 

1329 

1232 

12 

2441 

2316 

2194 

2076 

1961 

1849 

1739 

1633 

1528 

1427 

1327 

1230 

13 

2439 

2314 

2192 

2074 

1959 

1847 

1737 

1631 

1527 

1425 

1326 

1229 

14 

2437 

2312 

2190 

2072 

1957 

1845 

1736 

1629 

1525 

1423 

1324 

1227 

15 

2435 

2310 

2188 

2070 

1955 

1843 

1734 

1627 

1523 

1422 

1322 

1225 

16 

2433 

2308 

2186 

2068 

1953 

1841 

1732 

1626 

1522 

1420 

1321 

1224 

17 

2431 

2306 

2184 

2066 

1951 

1839 

1730 

1624 

1520 

1418 

1319 

1222 

18 

2429 

2304 

2182 

2064 

1950 

1838 

1728 

1622 

1518 

1417 

1317 

1221 

19 

2436 

2302 

2180 

2062 

1948 

1836 

1727 

1620 

1516 

1415 

1316 

1219 

20 

2424 

2300 

2178 

2061 

1946 

1834 

1725 

1619 

1515 

1413 

1314 

1217 

21 

2422 

2298 

2176 

2059 

1944 

1832 

1723 

1617 

1513 

1412 

1313 

1216 

22 

2420 

2296 

2174 

2057 

1942 

1830 

1721 

1615 

1511 

1410 

1311 

1214 

23 

2418 

2294 

2172 

2055 

1940 

1828 

1719 

1613 

1510 

1408 

1309 

1213 

24 

2416 

2291 

2170 

2053 

1938 

1827 

1718 

1612 

1508 

1407 

1308 

1211 

25 

2414 

2289 

2169 

2051 

1936 

1825 

1716 

1610 

1506 

1405 

1306 

1209 

26 

2412 

2287 

2167 

2049 

1934 

1823 

1714 

1608 

1504 

1403 

1304 

1208 

27 

2410 

2285 

2165 

2047 

1933 

1821 

1712 

1606 

1503 

1402 

1303 

1206 

28 

2408 

2283 

2163 

2045 

1931 

1819 

1711 

1605 

1501 

1400 

1301 

1205 

29 

2405 

2281 

2161 

2043 

1929 

1817 

1709 

1603 

1499 

1398 

1300 

1203 

30 

2403 

2279 

2159 

2041 

1927 

1816 

1707 

1601 

1498 

1397 

1298 

1201 

31 

2401 

2277 

2157 

2039 

1925 

1814 

1705 

1599 

1496 

1395 

1296 

1200 

32 

2399 

2275 

2155 

2037 

1923 

1812 

1703 

1598 

1494 

1393 

1295 

1198 

33 

2397 

2273 

2153 

2035 

1921 

1810 

1702 

1596 

1493 

1392 

1293 

1197 

34 

2395 

2271 

2151 

2033 

1919 

1808 

1700 

1594 

1491 

1390 

1291 

1195 

35 

2393 

2269 

2149 

2032 

1918 

1806 

1698 

1592 

1489 

1388 

1290 

1193 

36 

2391 

2267 

2147 

2030 

1916 

1805 

1696 

1591 

1487 

1387 

1288 

1192 

37 

2339 

2265 

2145 

2028 

1914 

1803 

1694 

1589 

1486 

1385 

1287 

1190 

38 

23S7 

2263 

2143 

2026 

1912 

1801 

1693 

1587 

1484 

1383 

1285 

1189 

39 

23S4 

2261 

2141 

2024 

1910 

1799 

1691 

1585 

1482 

1382 

1283 

1187 

40 

2332 

2259 

2139 

2022 

1908 

1797 

1G89 

1584 

1481 

1380 

1282 

1186 

41 

2330 

2257 

2137 

2020 

1906 

1795 

1687 

1582 

1479 

1378 

1280 

1184 

42 

2378 

2255 

2135 

2018 

1904 

1794 

1686 

1580 

1477 

1377 

1278 

1182 

43 

2376 

2253 

2133 

2016 

1903 

1792 

1684 

1578 

1476 

1375 

1277 

1181 

44 

2374 

2251 

2131 

2014 

1901 

1790 

1682 

1577 

1474 

1373 

1275 

1179 

45 

2372 

2249 

2129 

2012 

1899 

1788 

1680 

1575 

1472 

1372 

1274 

1178 

46 

2370 

2247 

2127 

2010 

1897 

1786 

1678 

1573 

1470 

1370 

1272 

1176 

47 

2368 

2245 

2125 

2009 

1895 

1785 

1677 

1571 

1469 

1368 

1270 

1174 

48 

2366 

2243 

2123 

2007 

1893 

1783 

1675 

1570 

1467 

1367 

1269 

1173 

49 

2364 

2241 

2121 

2005 

1891 

1781 

1673 

1568 

1465 

1365 

1267 

1171 

50 

2362 

2239 

2119 

2003 

1889 

1779 

1671 

1566 

1464 

1363 

1266 

1170 

51 

2359 

2237 

2117 

2001 

1888 

1777 

1670 

1565 

1462 

1362 

1264 

1168 

52 

2357 

2235 

2115 

1999 

1886 

1775 

1668 

1563 

1460 

1360 

1262 

1167 

53 

2355 

2233 

2113 

1997 

1884 

1774 

1666 

1561 

1459 

1359 

1261 

1165 

54 

2353 

2231 

2111 

1995 

1882 

1772 

1664 

1559 

1457 

1357 

1259 

1163 

55 

2351 

2229 

2109 

1993 

1880 

1770 

1663 

1558 

1455 

1355 

1257 

1162 

56 

2349 

2227 

2107 

1991 

1878 

1768 

1661 

1556 

1454 

1354 

1256 

1160 

57 

2347 

2225 

2105 

1989 

1876 

1766 

1659 

1554 

1452 

1352 

1254 

1159 

58 

2345 

2223 

2103 

1987 

1875 

1765 

1657 

1552 

1450 

1350 

1253 

1157 

59 

2343 

2220 

2101 

1986 

1873 

1763 

1655 

1551 

1449 

1349 

1251 

1156 

60 

2341 

2218 

2099  1 9841 

1871 

1761 

1654 

1549  1447) 

1347 

1249 

1154 

113 


TABLE  XCVI.     Logistical  Logarithyns, 


0 

46 

47 

48 
2880 
0969 

49   50  1  51  1 

52 

53 

54 

55 

56  j 
33601 

57 

_58 

59 

2760 
1154 

2820 
1061 

2940 

3000  3060 

3120 
0621 

3180 
0539 

3240 
0458 

3300 

0378 

3420 

3480 

3540 

7)880 

0792  0706  1 

0300 

0223 

0147 

0073 

1 

1152 

1059 

0968 

0878 

0790  0704 

0620 

0537 

0456 

0377 

0298 

0221 

014; 

0072 

2 

1151 

1057 

0966 

0877 

0789 

0703 

0619 

0536 

0455 

0375 

0297 

0220 

014^ 

0071 

3 

1149 

1056 

0965 

0875 

0787 

0702 

0617 

0535 

0454 

0374 

0296 

0219 

0143 

0069 

4 

1148 

1054 

0963 

0874 

0786 

0700 

0616 

0533 

0452 

0373 

0294 

0218 

014^ 

0068 

6 

1146 

1053 

0962 

0872 

0785 

0699 

0615 

0532 

0451 

0371 

0293 

0216 

0141 

0067 

6 

1145 

1051 

0960 

0871 

0783 

0697 

0613 

0531 

0450 

0370 

0292 

0215 

0140 

0066 

7 

1143 

1050 

0959 

0869 

0782 

0696 

0612 

0529 

0448 

0369 

0291 

0214 

0139 

0064 

8 

1141 

1048 

0957 

0868 

0780 

0694 

0610 

0528 

0447 

0367 

0289 

0213 

0137 

0063 

9 

1140 

1047 

0956 

0866 

0779 

0693 

0609 

0526 

0446 

0366 

0288 

0211 

0136 

0062 

10 

1138 

1045 

0954 

0865 

0777 

0692 

0608 

0525 

0444 

0365 

0287 

0210 

0135 

0061 

11 

1137 

1044 

0953 

0863 

0776 

0690 

0606 

0524 

0443 

0363 

0285 

0209 

0134 

0060 

12 

1135 

1042 

0951 

0862 

0774 

0689 

0605 

0522 

0442 

0362 

0284 

0208 

0132 

0058 

13 

1134 

1041 

0950 

0860  1  0773 

0687 

0603 

0521 

0440 

0361 

0283 

0206 

0131 

0057 

14 

1132 

1039 

0948 

0859  i  0772 

0680 

0602 

0520 

0439 

0359 

0282 

0205 

0130 

0056 

15 

1130 

1037 

0947 

0857  j  0770 

0685 

0601 

0518 

0438 

0358 

0280 

0204 

0129 

0O55 

16 

1129 

1036 

0945 

0856  j  0769 

0683 

0599 

0517 

0436 

0357 

0279 

0202 

0127 

0053 

17 

1127 

1034 

0944 

0855 1 0767 

0682 

0598 

0516 

0435 

0356 

0278 

0201 

0126 

0052 

18 

1126 

1033 

0942 

0853  0766 

0680 

0596 

0514 

0434 

0354 

0276 

0200 

0125 

0051 

19 

1124 

1031 

0941 

0852  0764 

0679 

0595 

0513 

0432 

0353 

0275 

0199 

0124 

0050 

20 

1123 

1030 

0939 

0850  0763 

0678 

0594 

0512 

0431 

0352 

0274 

0197 

0122 

0049 

21 

1121 

1028 

0938 

0849  I  0762 

0676 

0592 

0510 

0430 

0350 

0273 

0196 

0121 

0047 

22 

1119 

1027 

0936 

0847 1 0760 

0675 

0591 

0509 

0428 

0349 

0271 

0195 

0120 

0046 

23 

1118 

1025 

0935 

0846 1 0759 

0673 

0590 

0507 

0427 

0348 

0270 

0194 

0119 

0045 

24 

1116 

1024 

0933 

0844  ,  0757 

0672 

0588 

0506 

0426 

0346 

0269 

0192 

0117 

0044 

25 

1115 

1022 

0932 

0843  0756 

0670 

0587 

0505 

0424 

0345 

0267 

0191 

0116 

0042 

26 

1113 

1021 

0930 

0841  0754 

0669 

0585 

0503 

0423 

0344 

0266 

0190 

0115 

0041 

27 

1112 

1019 

0929 

0840  1  0753 

0668 

0584 

0502 

0422 

0342 

0265 

0189 

0114 

0040 

28 

1110 

1018 

0927 

0838  0751 

0666 

0583 

0501 

0420 

0341 

0264 

0187 

0112 

0039 

29 

1109 

1016 

0926 

0837 

0750 

0665 

0581 

0499 

0419 

0340 

0262 

0186 

0111 

0038 

30 

1107 

1015 

0924 

0835 

0749 

0663 

0580 

0498 

0418 

0339 

0261 

0185 

Olio 

0036 

31 

1105 

1013 

0923 

0834 

0747 

0662 

0579 

0497 

0416 

0337 

0260 

0184 

0109 

0035 

32 

1104 

1012 

0921 

0833 

0746 

0661 

0577 

0495 

0415 

0336 

0258 

0182 

OI07 

0034 

33 

1102 

1010 

0920 

0831 

0744 

0659 

0576 

0494 

0414 

0335 

0257 

0181 

3106 

0033 

34 

1101 

1008 

0918 

0830 1 0743 

0658 

0574 

0493 

0412 

0333 

0256 

0180 

01O5 

0031 

35 

1099 

1007 

0917 

0828  i  0741 

0656 

0573 

0491 

0411 

0332 

0255 

0179 

0104 

0030 

36 

1098 

1005 

0915 

0827  0740 

0655 

0572 

0490 

0410 

0331 

0253 

0177 

0103 

0029 

37 

1096 

1004 

0914 

0825  0739 

0654 

0570 

0489 

0408 

0329 

0252 

0176 

0101 

0028 

38 

1095 

1002 

0912 

0824  0737 

0652 

0569 

0487 

0407 

0328 

0251 

0175 

0100 

0027 

39 

1093 

1001 

0911 

0822  0736 

0651 

0568 

0486 

0406 

0327 

0250 

0174 

0099 

0025 

40 

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0172 

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41 

1090 

0998 

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0819  0733 

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0565 

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0171 

0096 

0023 

42 

1088 

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0818 ,0731 

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0563 

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0402 

0323 

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0170 

0095 

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43 

1087 

0995 

0905 

0816  0730 

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0480 

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0169 

0094 

0021 

44 

1085 

0993 

0903 

0815; 0729 

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0561 

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0167 

0093 

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45 

1084 

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46 

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47 

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48 

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0638 

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49 

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51 

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62 

1073 

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53 

1071 

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0001 

60 

1061 

0969  0880 

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0539 

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0378 

0300  0223 

0147 

0073 

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